Algebra Based Physics. Motion in One Dimension. 1D Kinematics Graphing Free Fall 2016.notebook. August 30, Table of Contents: Kinematics

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1 Table of Contents: Kinematics Algebra Based Physics Kinematics in One Dimension Motion in One Dimension Aerage Speed Position and Reference Frame Displacement Aerage Velocity Instantaneous Velocity Acceleration Kinematics Equation Free Fall Acceleration Due to Graity Kinematics Equation Kinematics Equation 3 Mixed Kinematics Problems Graphing Click on the topic to go to that section Jun 9 :3 PM Distance Motion in One Dimension We all know what the distance between two objects is... So what is it? What is distance? What is length? ALSO you can't use the words "distance" or "length" in your definition; that would be cheating. Jun 9 :7 PM Distance We'll be using meter as our standard for measuring distance. The symbol for distance is "d". And the unit for the meter is "m" Time Similarly, eeryone knows what time is... But try defining it; what is time? Remember you can't use the word "time" or an equialent to the word "time", in your definition. d = 0. m

2 Speed Speed is defined as the distance traeled diided by the time it took to trael that distance. Aerage Speed speed = distance time s = d t speed = meters/sec Speed is not a fundamental aspect of nature, it is the ratio of two things that are. Return to Table of Contents Jun 9 :7 PM Aerage Speed The speed we hae been calculating is a constant speed oer a short period of time. Another name for this is instantaneous speed. If a trip has multiple parts, each part must be treated separately. In this case, we can calculate the aerage speed for a total trip. Determine the aerage speed by finding the total distance you traeled and diiding that by the total time it took you to trael that distance. Distance and Time Interals In physics we use subscripts in order to aoid any confusion with different distances and time interals. For example: if an object makes a multiple trip that has three parts we present them as d, d, d3 and the corresponding time interals t, t, t3. Sep 8 4:0 PM Jul 3 0:5 AM Aerage Speed & Non Uniform Motion The following pattern of steps will help us to find the aerage speed: Find the total distance dtotal = d+ d+ d3 Find the total time ttotal = t + t + t3 Aerage Speed Example You ride your bike home from school by way of your friend s house. It takes you 7 minutes (40 s) to trael the 500 m to his house. You spend 0 minutes there, before traeling 3500 m to your house in 9 minutes (540 s). What was your aerage speed for this trip? Use the aerage speed formula sag = dtotal ttotal To keep things clear, we can use a table (graphic organizer) to keep track of the information... Jul 3 0:5 AM Sep 8 4:0 PM

3 Example Step Write the gien information in the table below: Example Step Write the gien information in the table below: Segment Distance Time Speed I II III Total /Ag. (m) (s) You ride your bike home from school by way of your friend s house. It takes you 7 minutes (40 s) to trael the 500 m to his house. You spend 0 minutes (600 s) there, before traeling 3500 m to your house in 9 minutes (540 s). What was your aerage speed for this trip? Segment Distance Time Speed I II III Total /Ag. (m) (s) You ride your bike home from school by way of your friend s house. It takes you 7 minutes (40 s) to trael the 500 m to his house. You spend 0 minutes (600 s) there, before traeling 3500 m to your house in 9 minutes (540 s). What was your aerage speed for this trip? Sep 8 4:0 PM Sep 8 4:0 PM Example Step Segment Distance Time Speed (m) (s) I II III Total /Ag. Write the gien information in the table below: You ride your bike home from school by way of your friend s house. It takes you 7 minutes (40 s) to trael the 500 m to his house. You spend 0 minutes (600 s) there, before traeling 3500 m to your house in 9 minutes (540 s). What was your aerage speed for this trip? Example Step Next, use the gien information to find the total distance and total time Segment Distance Time Speed Total /Ag. (m) (s) I II III You ride your bike home from school by way of your friend s house. It takes you 7 minutes (40 s) to trael the 500 m to his house. You spend 0 minutes (600 s) there, before traeling 3500 m to your house in 9 minutes (540 s). What was your aerage speed for this trip? dtotal = d+ d+ d3 Sep 8 4:0 PM Feb 7 3:58 PM Example Step Next, use the gien information to find the total distance and total time Example Step 3 Next use total distance and time to find aerage speed. Segment Distance Time Speed (m) (s) I II III Total /Ag You ride your bike home from school by way of your friend s house. It takes you 7 minutes (40 s) to trael the 500 m to his house. You spend 0 minutes (600 s) there, before traeling 3500 m to your house in 9 minutes (540 s). What was your aerage speed for this trip? Segment Distance Time Speed (m) (s) I II III Total /Ag You ride your bike home from school by way of your friend s house. It takes you 7 minutes (40 s) to trael the 500 m to his house. You spend 0 minutes (600 s) there, before traeling 3500 m to your house in 9 minutes (540 s). What was your aerage speed for this trip? ttotal = t + t + t3 s ag = d total t total Feb 7 4:04 PM Feb 7 4:09 PM 3

4 Example Solution Next use total distance and time to find aerage speed. Segment Distance Time Speed (m) (s) I II III Total /Ag s ag = You ride your bike home from school by way of your friend s house. It takes you 7 minutes (40 s) to trael the 500 m to his house. You spend 0 minutes (600 s) there, before traeling 3500 m to your house in 9 minutes (540 s). What was your aerage speed for this trip? d total t total = 6000 m 560 s = Position and Reference Frames Return to Table of Contents Sep 8 4:0 PM Jun 9 :7 PM Position and Reference Frames Speed, distance and time didn't require us to define where we started and where we ended up. They just measure how far we traeled and how long it took to trael that far. Howeer, much of physics is about knowing where something is and how its position changes with time. Position and Reference Frames A reference frame lets us define where an object is located, relatie to other objects. For instance, we can use a map to compare the location of different cities, or a globe to compare the location of different continents. To define position we hae to use a reference frame. Howeer, not eery reference frame is appropriate for eery problem. MOTION IS RELATIVE. THIS MEANS THAT HOW YOU MEASURE SOMEONE ELSE"S MOTION DEPENDS ON WHERE YOU ARE STANDING AND YOUR OWN MOTION AS COMPARED TO SOMETHING OR SOMEONE ELSE WALKING DEMO Who is obsering matters! What is motion? *Motion is a change in an object's position relatie to a gien obserer during a certain change in time. *Without identifying the obserer, it is impossible to say whether the object of interest moed. *Physicists say motion is relatie, meaning that the motion of any object of interest depends on the point of iew of the obserer. Identify the object of interest AND the Obserer An obserer in a spaceship describes the motion of the Sun differently than an obserer standing on Earth. The obserer must be specified! MOTION IS RELATIVE. THIS MEANS THAT HOW YOU MEASURE SOMEONE ELSE"S MOTION DEPENDS ON WHERE YOU ARE STANDING AND YOUR OWN MOTION AS COMPARED TO SOMETHING OR SOMEONE ELSE WALKING DEMO 4

5 Obserational experiment Reference frames require: An object of reference (or a point on an object if the object is large) A coordinate system with a scale for measuring distance A clock to measure time Different obserers can describe the same process differently, including whether motion is een occurring. Reference Frame Actiity Results Reference Frames In this course, we'll usually: Define the origin as a location labeled "zero" Create three perpendicular axes : x, y and z for direction Use the meter as our unit of measure In this course, we will be soling problems in one dimension (D) and two dimensions (D). We will start with D motion You'll probably find that you need a starting point (an origin) Typically, we use the x axis for D direction. A set of directions (for instance left right, forward backward, updown) +x will usually be to the right A unit of measure (how far to go in each direction) x would then be to the left x +x Usually we'll: The Axis Define We could the define origin as it the a location opposite labeled way, "zero" but unless specified otherwise, this is what we'll assume. We also can think about Three compass axes: directions x, y and in z terms for direction of positie and negatie. For example, North would be positie and South negatie. The meter as our unit of measure The symbol for D position is normally "x" or in the case of your book "d" (book uses "d" because it is not associated with the x y coordinate system). Linear motion Linear motion is a model of motion that assumes that an object, considered as a point like object, moes along a straight line. A car moing along a straight highway can be modeled with linear motion; we simplify the car as a point, which is small compared to the length of the road. A tire of the car cannot be modeled with linear motion. 5

6 Motion diagrams contain: Patterns found from motion diagrams Dots representing the location of the object for progressie, equal time interals Velocity ectors on each dot representing the elocity of the object at each time interal (the length of the elocity ector represents how fast the object is moing) Velocity change arrows showing how the elocity ectors are changing The spacing of the dots allows us to isualize motion. When the object traels without speeding up or slowing down, the dots are eenly spaced. When the object slows down, the dots get closer together. When the object moes faster and faster, the dots get farther apart. The specified location of the obserer Velocity change arrows Finding elocity change arrows Delta means "change in." Change always means final (f) minus initial (o): what it is now compared to what it was before. Consider two adjacent elocity ectors, in this example at points and 3. The arrow aboe reminds us that elocity is a ector; it has both magnitude and direction. Find which ector would need to be added to the elocity corresponding to point to get the elocity corresponding to point 3. Constructing a motion diagram Quantities for describing motion Motion diagrams represent motion qualitatiely. To analyze situations, we need to describe motion quantitatiely. These quantities are needed to describe linear motion: > Time and time interal > Position, displacement, distance, and path length > Scalar component of displacement for motion along one axis 6

7 Time and time interal Position, displacement, distance, and path length The time t is a clock reading. The time interal (t t) or Δt is a difference in clock readings. (The symbol delta represents "change in" and is the final alue minus the initial alue.) These are both scalar quantities. The SI units for both quantities are seconds (s). These quantities describe the location and motion of an object. Position is an object's location with respect to the origin (Where you start) Displacement is a ector that starts from an object's initial position and ends at its final position (it's the oerall change in position). Distance is a scalar and represents how much ground an object has coered or how far it has gone during the time period analyzed Path length (or total distance traeled) is how far the object moed as it traeled from its initial position to its final position. > Imagine laying a string along the path the object took. The length of the string is the path length/total distance traeled. Run Logger Pro Videos Constant Car Lab Problem: Determine a method to graph motion Materials: Meter stick, stop watch, constant elocity cars with different battery combination. VIdeo Camera Procedure: >. 5 students at uniform interals from to 3 meters that corresponds to about sec interals. Start the car. >. When car passes the starting line, shout GO. All timers start stop watches. Start the ideo. > 3. When car passes a timer at designated time mark the distance ( sec interals). > 4. Record data within the data table determined in class. > 5. Repeat two more times (only ideo one of each runs) > 6. Change out the car and perform steps 5 for nd car. Constant Car Lab All Groups Make a table for each of your runs. Determine the aerage time for each distance. Make a motion diagram on your whiteboard for both cars. Graph nd car directly below st car. Proide significant figures as appropriate. YOU MUST USE AVG VALUES. WE ARE NOW IN THE LAB...RAW DATA IS NOT MUCH GOOD BECAUSE OF EXEPRIEMNTAL ERROR THAT AFFECT PRECISION AND ACURACY. Graph both in Logger Pro (remember the ideo I just played) Discussion Topics...Be ready To : What is different between the two graphs What is the graph showing. Can we draw displacement and elocity ectors on the particle motion plot? Why isn t the distance between the dots exactly the same between each time interal. What is the distance trael for both cars. Lets graph Distance s Time for both cars (must determine dependant and independent ariables). Determine the slope and tell me what it represents Can we determine precision and/or accuracy with our experiment. If so make it happen!. Displacement Displacement Now that we understand how to define position and distance, we can talk about a change in position; a displacement. The symbol for "change" is the Greek letter "delta" "Δ". So "Δx" means the change in x or the change in position Displacement is a Vector!!!!!!! Return to Table of Contents Jun 9 :7 PM 7

8 Displacement +y Displacement +y Displacement describes how far you are from where you started, regardless of how you got there. x y +x For instance, if you drie 60 miles from Pennsylania to New Jersey... x y x0 (In physics, we label the starting position x0) +x and then 0 miles back toward Pennsylania. Displacement +y You hae traeled: Displacement +y x y x0 xf +x (We also label the final position xf ) a distance of 80 miles, and a displacement of 40 miles, since that is how far you are from where you started x x0 xf +x y we can calculate displacement with the following formula: Δx = X f X o Displacement Displacement Howeer, displacement can be positie or negatie since you can end up to the right or left of where you started. Distance (the scalar) can only be positie alues since it is impossible to trael a negatie distance. +y +y x x o x f +x x x f x o +x Imagine trying to measure a negatie length with a meter stick... y Displacement is positie. y Displacement is negatie. 8

9 Vectors and Scalars Scalar a quantity that has only a magnitude (number or alue) Vector a quantity that has both a magnitude and a direction How far your ending point is from your starting point is known as: A distance B displacement Which of the following are ectors? Scalars? Quantity Vector Scalar C a positie integer D a negatie integer Time Distance Displacement Speed Sep :6 PM May 9 9:35 PM Starting from the origin, a car traels 4km east and then 7 km west. What is the total distance traeled? A 3 km B 3 km C 7 km D km 3 Starting from the origin, a car traels 4km east and then 7 km west. What is the net displacement from the original point? A 3 km west B 3 km east C 7 km west D km east 9A May 9 9:38 PM May 9 9:38 PM Kinematics Graphs (D motion/linear motion): Correspondence between a motion diagram and position ersus time graph Graph : Position Time Graph (P T Graph) Kinematics graphs can contain more precise information than motion diagrams. Time t is usually the independent ariable: the horizontal axis. Position x is the dependent ariable (position changes with time): the ertical axis (een if the motion is horizontal!). A trendline is a best fit cure that passes as close as possible to the data points. The position of each dot on the motion diagram corresponds to a point on the position axis. The graph line combines information about the position of an object and the clock reading when this position occurred. 9

10 Mathematics of linear motion A dependent ariable, usually y, depends on the alue of an independent ariable, usually x. y(x) = f(x) is an operation that one needs to do if x is an input and y is the output. For a straight line, y(x) = kx + b, where k is the slope and b is the y intercept; the alue of y when x = 0. For motion along the x axis, we write x(t) to depict that the motion is dependent on time; x(t) = kt + b. Jun 4 0:38 AM Connecting graphical representations of linear motion to a mathematical representation Velocity: Slope of the position ersus time graph A linear function is written: x(t) = kt + b k is the slope of the line; it is the change in the dependent ariable diided by the change in the independent ariable. The slope k can be found from k has units of m/s and indicates how the position changes with time. k can be positie or negatie; it represents not only how fast, but also in which direction an object is moing. If the slope is positie, the object is moing along the +x axis. If the slope is negatie, the object is moing along the x axis. The magnitude of the slope (which is always positie) is the speed of the object. The speed and the direction together are called the elocity of the object. b is the location of the object at t = 0; it is x0. Position Time Graphs Graphing Problems Two different runners At what time do A and B hae the same position? Particle model? Return to Table of Contents graphing Jun 4 0:38 AM 0

11 Jun 4 0:38 AM Jun 4 0:38 AM Starting at the position, x 0 = 4 m, you trael at a constant elocity of + m/s for 6s. a. Determine your position at the times of 0s; s; 5s; and 6s. Starting at the position, x 0 = 4 m, you trael at a constant elocity of + m/s for 6s. b. Draw the Position ersus Time for your trael during this time. Drag and drop the data points on the graph in order to construct the s t pattern! x (m) Draw a line of best fit to obsere the pattern Jul 8:37 AM Jul 8:38 AM Starting at the position, x 0 = 0 m, you trael at a constant elocity of m/s for 6s. d. Draw the Position ersus Time for your trael during this time. Drag and drop the data points on the graph in order to construct the s t pattern! x (m) Analyzing Position s Time Graphs Recall earlier in this unit that slope was used to describe motion. The slope in a position s. time graph is Δx/Δt, which is equal to elocity. x (m) Δt Δx = Δx/Δt Draw a line of best fit to obsere the pattern Therefore, slope is equal to elocity on a position s. time graph. Feb 8 :0 AM Jul 3 :07 PM

12 Analyzing Position s Time Graphs Equation of motion for constant elocity linear motion A positie slope is a positie elocity, a negatie slope is a negatie elocity, and a slope of zero means zero elocity. positie slope > 0 negatie slope < 0 zero slope = 0 Couple of things: x (m) x (m) x (m). This equation elocity is constant so is the same throughout the problem. Remember is a ector (I am using bold instead of a top arrow because smartboard does not hae that ability). You hae used this equation before in Math: D=Rate*time D=rt A positie elocity means moing in the positie direction, a negatie elocity means moing in the negatie direction, and zero elocity means not moing at all. 3. Physics: x =elocity*time or xt (x is the elocity in the x direction) Note: book sometimes uses d instead of x and therefore does not hae a subscript. Jul 3 :07 PM Aerage Velocity Aerage Velocity Speed is defined as the ratio of distance and time. Note Ag Speed and Distance Traeled are SCALERS Aerage speed = distance traeled time elapsed s = d t Similarly, elocity is defined as the ratio of displacement and time NOTE: Ag elcoity and displacement are VECTORS!!! Return to Table of Contents Aerage elocity = displacement time elapsed Δx = Δt Jun 9 :7 PM Aerage Velocity Speeds are always positie, since speed is the ratio of distance and time; both of which are always positie. 4 You trael 60 meters to the left in 0 s and then you trael 60 meters to the right in 30 s; what is your aerage elocity? Aerage speed = distance traeled time elapsed s = d t But elocity can be positie or negatie, since elocity is the ratio of displacement and time; and displacement can be negatie or positie. Aerage elocity = displacement time elapsed Δx = Δt Usually, right is positie and left is negatie.

13 D Kinematics Graphing Free Fall 06.notebook 5 You trael 60 meters to the left in 0 s and then you trael 60 meters to the right in 30 s what is your aerage speed? Instantaneous Velocity Return to Table of Contents Jun 9 :7 PM Instantaneous Velocity Instantaneous Velocity Sometimes the aerage elocity is all we need to know about an object's motion. Aerage elocity is defined as change in position oer time. This tells us the 'aerage' elocity for a gien length or span of time. For example: A race along a straight line is really a competition to see whose aerage elocity is the greatest. If we want to know the speed or elocity of an object at a specific point in time (with this radar gun for example), we want to know the instantaneous elocity... The prize goes to the competitor who can coer the displacement in the shortest time interal. But the aerage elocity of a moing object can't tell us how fast the object moes at any gien point during the interal Δt. Watch what happens when we look for the instantaneous elocity by reducing the amount of time we take to measure displacement. Radar Gun Demo!!!! Jul 3 0:58 AM Aug :5 PM Instantaneous Velocity Displacement Time 00m 0 s Instantaneous Velocity Velocity In an experiment, an object traels at a constant elocity. Find the magnitude of the elocity using the data aboe. Displacement Time Velocity 00m 0 s 0 m/s 0 m s What happens if we measure the distance traeled in the same experiment for only one second? What is the elocity? Aug :5 PM Aug :5 PM 3

14 Instantaneous Velocity Displacement Time Velocity What happens if we measure the distance traeled in the same experiment for a really small time interal? What is the elocity? 00m 0 s 0 m/s 0 m s 0 m/s 0.00m s Instantaneous Velocity Displacement Time Velocity 00 m 0 s 0 m/s 0 m s 0 m/s.0 m 0.0 s 0 m/s 0.0 m 0.00 s 0 m/s 0.00 m s 0 m/s m s 0 m/s m s 0 m/s Since we need time to measure elocity, we can't know the exact elocity "at" a particular time... but if we imagine a really small alue of time and the distance traeled, we can estimate the instantaneous elocity. Aug :5 PM Aug :5 PM Instantaneous Velocity Instantaneous Velocity In physics an instant has no duration at all; it refers to a single alue of time. One of the most common examples we can use to understand instantaneous elocity is driing a car and taking a quick look on the speedometer. At this point, we see the instantaneous alue of the elocity. The instantaneous elocity is the same as the magnitude of the aerage elocity as the time interal becomes ery ery short. Δx = Δt as Δt 0 Jul 3 :43 AM Aerage s. Instantaneous? Graphing Velocity Graph : The V T Kinematics Graph Jun 4 0:38 AM 4

15 Velocity Graphing Actiity Velocity Graphing Actiity The graph below shows elocity ersus time. The graph below shows elocity ersus time. How do you know the elocity is constant? When is the elocity increasing? Decreasing? Constant? Discuss. Jun 9 :0 PM Jun 9 :0 PM Velocity Graphing Actiity Velocity Graphing Actiity Use the graph to determine the Aerage Velocity of (b) b.) Use the graph to determine the Aerage Velocity of (a). a.) from 0s to s from s to 3s from 3s to 4s from 4s to 5s from 3s to 5s Jun 9 :03 PM Jun 9 :03 PM Velocity Graphing Actiity a.) 4 Use the graph to determine the 3 Instantaneous Velocity of (a) at seconds 4 6 Velocity Graphing Actiity a.) b.) 4 3 Use the graph to determine the Instantaneous Velocity of (b) at seconds b.) Jun 9 :03 PM Jun 9 :03 PM 5

16 Instantaneous Velocity These graphs show (a) constant elocity and (b) arying elocity. (a) When the elocity of a moing object is a constant the instantaneous elocity is the same as the aerage. Starting at the position, x 0 = 4 m, you trael at a constant elocity of + m/s for 6s. c. Draw the Velocity ersus Time graph for your trip. Drag and drop the data points on the graph in order to construct the s t pattern! (b) When the elocity of a moing object changes its instantaneous elocity is different from the aerage elocity. Draw a line of best fit to obsere the pattern. Jul 8:38 AM Finding displacement from a elocity graph Displacement x x0 between t0 = 0 and time t is the area under the V T graph. Remember = So looking at the V T graph: Finding displacement from a elocity graph Displacement can be negatie Area under the cure IS NOT Path traeled or total distance!!!! > Area is width times height = x(t t0) Since x= (x x0)/(t t0), (x x0) = x(t t0) When elocity is not constant On a elocity ersus time graph, elocity will be a straight line only if it is constant. Instantaneous elocity is the elocity of an object at a particular time. Acceleration characterizes the rate at which the elocity of an object is changing The simplest type of linear motion with changing elocity occurs when the elocity of the object increases or decreases by the same amount during the same time interal (a constant rate of change). In this simple case, the elocity ersus time graph will be linear. Aerage elocity is the ratio of the change in position and the time interal during which this change occurred. For motion at constant elocity, the instantaneous and aerage elocities are equal; for motion with changing elocity, they are not. 6

17 Finding acceleration from a elocity ersus time graph Acceleration Acceleration is the slope of the elocity ersus time graph: A larger slope indicates the elocity is increasing at a faster rate. Velocity is a ector quantity; therefore acceleration is also a ector quantity. The aerage acceleration of an object during a time interal is Acceleration Acceleration Return to Table of Contents You can feel a difference between uniform (constant) and nonuniform (accelerated) motion. When you moe in a nonuniform motion, you feel pushed or pulled. In contrast, when you are in uniform motion and your eyes are closed, you feel as though you are not moing at all. Jun 9 :7 PM Acceleration So far, we e talked about how far and how fast now we want to study how fast an object is getting faster The rate at which an object s elocity changes is called acceleration. Acceleration is measured in m/s/s or m/s. What is meant by per Displaying Acceleration on a Motion Diagram To determine the length and direction of an aerage acceleration ector, subtract two consecutie elocity ectors, as shown below. 7

18 Displaying Acceleration on a Motion Diagram Finding the Acceleration Vector Direction: You will hae: = f i = f + ( i) Acceleration Vector Always the same as Magnitude: Then diide by the time interal, t. Remember: Text: p. 69 Position, Velocity, Acceleration Positie and Negatie Acceleration When the object s acceleration is in the same direction as its elocity, the object s elocity increases. Note: Constant Velocity =No Change in elocity ACCELERATION = 0 When they are in opposite directions, the elocity decreases. So.. Positie and Negatie Acceleration Velocity and Acceleration 8

19 3 Ways to Show Acceleration Acceleration Acceleration is the rate of change of elocity. a = Δ Δt acceleration = change of elocity elapsed time Acceleration Units for Acceleration Acceleration is a ector, although in one dimensional motion we only need the sign. Since only constant acceleration will be considered in this course, there is no need to differentiate between aerage and instantaneous acceleration. a = Δ Δt m/s s = m/s When is acceleration negatie? Determining the elocity change from the acceleration Acceleration can be positie or negatie. If an object moing in the positie direction is slowing down, its elocity ersus time graph has a negatie slope, corresponding to a decreasing speed and negatie acceleration. An object can hae a negatie acceleration and speed up! Consider an object moing in the negatie direction with a negatie component of elocity, but whose speed is increasing in magnitude. From the slope of the elocity ersus time graphs, we see: ax = (x 0x)/(t t0), therefore: ax (t t0) = (x 0x) Set t0 = 0 to simplify (the clock starts at 0) and moe 0x to the other side: x = 0x + axt 9

20 Position as a function of time Motion at Constant Acceleration Graphical Approach If the area under the graph is length x width (A = lw), then: A = lw Δx A = 0t Since we know that =, t then area is really Δx. A = Δx = 0t The equation for displacement can be found from the area between the elocity ersus time graph line and the time axis. Motion at Constant Acceleration Graphical Approach Motion at Constant Acceleration Graphical Approach A = ½bh If the area under this graph is ½ base x height, then: Δ A = ½tΔ Since we know that a =, t Δ = at. ½at 0t Therefore, the area under a elocity s. time graph is displacement. It can be calculated by combining the preious two results. A = Δx = 0t + ½at A = Δx = ½t(at) = ½at x x 0 = 0t + ½at x = x0 + 0t + ½at Position of an object during linear motion with constant acceleration Graph of position ersus time for constant acceleration motion Position is quadratic in time (there is a t term), so the graph is parabolic. The slopes of the tangent lines (indicating the instantaneous elocity) are different for different times. 0

21 Graphing Acceleration Graph 3: Accleration s Time Graph Board Demos: Putting All The Graphs Together!!!! PLEASE NOTE: In this class acceleration will be 0 or constant!!! You need calculus to deal with an acceleration that changes with time. Question Which of the following statements correctly define acceleration? Acceleration is the rate of change of displacement of an object. Acceleration is the rate of change of elocity of an object. Acceleration is the amount of distance coered in unit time. Acceleration is the rate of change of elocity of an object. Jul 30 :4 AM Question What happens when the elocity ector and the acceleration ector of an object in motion are in same direction? A. The acceleration of the object increases. B. The elocity of the object increases. C. The object comes to rest. D. The elocity of the object decreases. Question 3 6 A horse gallops with a constant acceleration of 3m/s. Which statement below is true? A The horse's elocity doesn't change. B The horse moes 3m eery second. C The horse's elocity increases 3m eery second. D The horse's elocity increases 3m/s eery second. May 9 9:55 PM The Kinematic Equations of Motion or The FAB FOUR (acceleration is constant) Soling Problems After you read the problem carefully:. Draw a diagram (include coordinate axes).. List the gien information. 3. Identify the unknown (what is the question asking?) 4. Choose a formula (or formulas to combine) (Plug into 3 get 4) 5. Rearrange the equations to isolate the unknown ariable. 6. Substitute the alues and sole! 7. Check your work. (You can do the same operations to the units to check your work... try it!)

22 Motion at Constant Acceleration Kinematics Equation If elocity is changing at a constant rate, the aerage elocity is just the aerage of the initial and final elocities. = + o Δx = ( + o) t And we learned earlier that Some problems can be soled most easily by using these two equations together. = Δx t Δx = = Δx t + o ( + o) t Return to Table of Contents nl3c Jun 9 :7 PM 7 Starting from rest you accelerate to 0 m/s in 4.0s. What is your aerage elocity? Kinematics Equation = o + at Return to Table of Contents Sep 6 9:59 PM Jun 9 :7 PM Motion at Constant Acceleration a = Δ but since "Δ" means change Δt Δ = o and 8 Starting from rest, you accelerate at 8.0 m/s for 9.0s. What is your final elocity? a = o t at = o o = at Δt = t to if we always let to = 0, Δt = t Soling for "" = o + at This equation tells us how an object's elocity changes as a function of time. 0 Aug 4:30 PM

23 9 You hae an initial elocity of 5.0 m/s. You then experience an acceleration of.5 m/s for 4.0s; what is your final elocity? Motion at Constant Acceleration The graph on the right has a slope of Δ/Δt, which is equal to acceleration. Therefore, the slope of a elocity s. time graph is equal to acceleration. (slope) m =Δy/Δx (slope of elocity s. time) a =Δ/Δt Jul 3 :07 PM 0 The elocity as a function of time is presented by the graph. What is the acceleration? The elocity as a function of time is presented by the graph. Find the acceleration. TAxlgs Jul 3 3:49 PM Jul 4 9:59 AM Motion at Constant Acceleration The acceleration graph as a function of time can be used to find the elocity of a moing object. When the acceleration is constant the elocity is changing by the same amount each second. This can be shown on the graph as a straight horizontal line. In order to find the change in elocity for a certain limit of time we need to calculate the area under the acceleration line that is limited by the time interal. Motion at Constant Acceleration The change in elocity during first seconds is equialent to the shadowed area (4m x s = 48m). s The change in elocity during first seconds is 48 m/s. s Jul 4 0:07 AM Jul 4 0:07 AM 3

24 The following graph shows acceleration as a function of time of a moing object. What is the change in elocity during first 0 seconds? Kinematics Equation 3 x = x o + ot + ½at Return to Table of Contents Jul 4 0:33 AM Jun 9 :7 PM Δx = t Δx = t Motion at Constant Acceleration = + o = o + at 3 An airplane starts from rest and accelerates at a constant rate of 3.0 m/s for 30.0 s before leaing the ground. How far did it moe along the runway? x x o = ½ ( + o)t x x o = ½t + ½ ot x = x o + ½ ot + ½t x = x o + ½ ot + ½( o + at)t x = x o + ½ ot + ½ ot + ½at x = x o + ot + ½at We can combine these three equations to derie an equation which will directly tell us the position of an object as a function of time. Sep 3 8:03 PM 4 A Volkswagen Beetle moes at an initial elocity of m/s. It coasts up a hill with a constant acceleration of.6 m/s. How far has it traeled after 6.0 seconds? 5 A train pulling out of Grand Central Station accelerates from rest at a constant rate. It coers 800 meters in 0 seconds. What is its rate of acceleration? RG_fY Sep 3 8:03 PM Sep 3 8:09 PM 4

25 6 A Greyhound bus traeling at a constant elocity starts to accelerate at a constant.0 m/s. If the bus traels 500 meters in 0 seconds, what was its initial elocity? Kinematics Equation 4 Return to Table of Contents AiM Sep 3 8: PM Jun 9 :7 PM (+o)/ Motion at Constant Acceleration We can also combine these equations so as to eliminate t: 7 You accelerate from 0m/s to 60m/s while traeling a distance of 00m; what was your acceleration? Sep 6 0:4 PM 8 Beginning with a elocity of 5m/s, you accelerate at a rate of.0m/s. During that acceleration you trael 00m; what is your final elocity? 9 A ball with an initial elocity of 5m/s is subject to an acceleration of 9.8 m/s ; how high does it go before coming to a momentary stop? Sep 6 0:47 PM Sep 6 0:4 PM 5

26 0 An arrow is projected by a bow ertically up with a elocity of 40 m/s, and reaches a target in 3 s. How high is the target located? Mixed Kinematics Problems Return to Table of Contents Mixed Feb 7 7:0 PM An object accelerates from rest, with a constant acceleration of 8.4 m/s, what will its elocity be after s? An object accelerates from rest to a elocity of 34 m/s oer a distance of 70 m. What was its acceleration? Feb 7 6:59 PM Feb 7 6:57 PM The position ersus time graph, below, describes the motion of three different cars moing along the x axis. The position ersus time graph, below, describes the motion of three different cars moing along the x axis. a. Describe, in words, the elocity of each of the cars. Make sure you discuss each car s speed and direction. Position (m) Time (s) b. Calculate the elocity of each of the cars. Position (m) Time (s) Sep 6 0:4 PM Sep 6 0:4 PM 6

27 c. Draw, on one set of axes, the Velocity ersus Time graph for each of the three cars. Position (m) Time (s) The elocity s time graph, below, describes the motion of an object moing along the x axis. When is elocity zero? Jul 8:53 AM Feb 7 7:9 PM The elocity s time graph, below, describes the motion of an object moing along the x axis. 3 The elocity s time graph, below, describes the motion of an object moing along the x axis Describe in words what is happening to the speed during the following interals. a) 0s to s b) s to 3s c) 3s to 4 sec d) 4s to 5s e) 5s to 6s Determine the aerage speed during the following interals. a) 0s to s b) s to 3s c) 3s to 4 sec d) 4s to 5s e) 5s to 6s f) 3s to 5s Feb 6 7:0 PM Jun 8 4:54 PM The elocity s time graph, below, describes the motion of an object moing along the x axis. The elocity s time graph, below, describes the motion of an object moing along the x axis Vag = (Vf + Vi)/ a) 0s to s Vag = +m/s b) s to 3s Vag = +4m/s c) 3s to 4s Vag = +m/s d) 4s to 5s Vag = m/s e) 5s to 6s Vag = 4m/s f) 3s to 5s Vag = 0m/s Determine the displacement during the following interals. a) 0s to s b) s to 3s c) 3s to 4 sec d) 4s to 5s e) 5s to 6s 4 Determine the net displacement during the first four seconds of trael. Feb 6 7:0 PM Feb 7 7:5 PM 7

28 Determining Acceleration from a t Graph Graphs A, B, C, D, and E, as shown on the right, represent the motions of fie different runners. Assume that the positie direction has been chosen to be east. What are the accelerations for each. Determining Acceleration from a t Graph The slopes of Graphs A and E are zero. Thus, the accelerations are zero. Both Graphs A and E show motion at a constant elocity Graph A to the east and Graph E to the west. Graph B shows motion with a positie elocity. The slope of this graph indicates a constant, positie acceleration. Determining Acceleration from a t Graph Graph C has a negatie slope, showing motion that begins with a positie elocity, slows down, and then stops. This means that the acceleration and elocity are in opposite directions. The point at which Graphs C and B cross shows that the runners elocities are equal at that point. It does not, howeer, gie any information about the runners positions.. Determining Acceleration from a t Graph Graph D indicates moement that starts out toward the west, slows down, and for an instant gets to zero elocity, and then moes east with increasing speed. Determining Acceleration from a t Graph The slope of Graph D is positie. Because the elocity and acceleration are in opposite directions, the elocity decreases and equals zero at the time the graph crosses the axis. After that time, the elocity and acceleration are in the same direction and the elocity increases. Summary Kinematics is the description of how objects moe with respect to a defined reference frame. Displacement is the change in position of an object. Aerage speed is the distance traeled diided by the time it took; aerage elocity is the displacement diided by the time. 8

29 Summary (continued) Instantaneous elocity is the limit as the time becomes infinitesimally short. Aerage acceleration is the change in elocity diided by the time. Instantaneous acceleration is the limit as the time interal becomes infinitesimally small. Summary (continued) There are four equations of motion for constant acceleration, each requires a different set of quantities. = o + at x = x o + o t + ½at = o + a(x x o) = + o Feb 7 :3 AM Describing Motion with Graphs Plot and interpret a distance time graph and a elocity time graph. Deduce from the shape of a distance time graph when a body is: (a) at rest (b) moing with uniform elocity (c) moing with non uniform elocity 3. Deduce from the shape of a elocity time graph when a body is: (a) at rest (b) moing with uniform elocity (c) moing with uniform acceleration (d) moing with non uniform acceleration Key Concepts Position time Graph. Slope of the Position time Graph is the elocity of the moing object Velocity time Graph Slope of the Velocity time Graph is the acceleration of the moing object.. Area under the Velocity time Graph is the distance traelled. 4. Calculate the area under a elocity time graph to determine the distance traelled for motion with uniform elocity or uniform acceleration. Free fall Free Fall: Acceleration Due to Graity Or Graity is Making me Fall Return to Table of Contents The hypothetical motion of falling objects in the absence of air is a model of the real process and is called free fall. Galileo hypothesized, based on experiments, that free fall occurs exactly the same way for all objects regardless of their mass and shape. Galileo hypothesized that speed increases in the simplest way linearly with time of flight; the acceleration of freefalling objects is constant. Jun 9 :7 PM 9

30 Acceleration Due to Graity Acceleration Due to Graity So.. Galileo concluded: neglecting the effect of the air, all objects in free fall had the same acceleration. It didn t matter what they were made of, how much they weighed, what height they were dropped from, or whether they were dropped or thrown. The acceleration of falling objects, gien a special symbol, g, is equal to 9.80 m/s. We can use 0 m/s in most of our calculations. The acceleration due to graity is the acceleration of an object in free fall. Jun 5 0:8 AM Object is in Free Fall when: > The only force acting on it is the force of graity > Its not touching other objects > There is no air resistance (The acuum that you can breath) When on earth an object is in Free Fall > a y = g = 9.8m/s (or 0 m/s for calculations) > Not on earth little g is different!!!!! g mercury = 3.6m/s g moon =.6 m/s (approx /6 of earth) g enus = 8.83 m/s g mars = 3.75m/s Jun 5 0:8 AM Acceleration Due to Graity g because acceleration is down.its in the negatie y direction Howeer acceleration due to graity (g) is positie!!!! Its only when we talk about it in an xy system we say negatie g aries from place to place on earth Howeer, for our problems we are going to assume a constant g so we can use our Uniform Accelerated Motion equations (kinematic equation). Pick a g (most cases 9.8/0.0 m/s ) and stick with it (Global s local...specific location) If eerything accelerates at the same rate, does that mean eerything falls at the same rate? Een they hae a different weight? Yes Een if they are different sizes and shapes? Yes Howeer, remember its YES as long as we do not include air resistance ( for example a feather falls at the same rate but is ery affected by air resistance where the lacrosse ball is affected ery little. Little g is actually not called graity. g is the free fall acceleration. Jun 5 0:5 AM Jun 5 0:5 AM Motion diagrams and graphs for free fall Free Fall All unsupported objects fall towards Earth with the same acceleration. We call this acceleration the "acceleration due to graity" and it is denoted by g. g = 9.8 m/s Keep in mind, ALL objects accelerate towards the earth at the same rate. g is a constant! Click here to watch Galileo's famous experiment performed on the moon Sep 5 0: AM 30

31 Free Fall It stops momentarily. What = happens 0 at the top? g = 9.8 m/s Free Fall s It stops momentarily. = 0 g = 9.8 m/s It What slows happens down. when it (negatie goes up? acceleration) g = 9.8 m/s It speeds up What happens when it (negatie acceleration) goes down? g = 9.8 m/s It slows down. (negatie acceleration) g = 9.8 m/s It speeds up. (negatie acceleration) g = 9.8 m/s An object is thrown upward with initial elocity, o (Click on question for answer.) It What returns happens with when its it original lands? elocity. An object is thrown upward with initial elocity, o It returns with its original elocity. Sep :0 PM Jun 9 :3 PM On the way up: 0 a a a a t = 3 s t = s t = s t = 0 s Free Fall On the way down: 0 a a a t = 0 s t = s t = s t = 3 s For any object thrown straight up into the air, this is what the elocity s. time graph looks like. Free Fall An object is thrown upward with initial elocity, o It stops momentarily. = 0 g = 9.8 m/s It returns with its original elocity but in the opposite direction. Jun 30 :38 AM Jun 30 :07 PM Free Fall Question A ball is tossed straight up in the air. At its ery highest point, the ball s instantaneous acceleration ay is Positie. Negatie. Zero. The figure shows the motion diagram for an object that was released from rest and falls freely. The diagram and the graph would be the same for all falling objects. 05 Pearson Education, Inc. Jun 5 0:5 AM Jun 5 0:5 AM 3

32 Question An arrow is launched ertically upward. It moes straight up to a maximum height, then falls to the ground. The trajectory of the arrow is noted. At which point of the trajectory is the arrow s acceleration the greatest? The least? Ignore air resistance; the only force acting is graity. Question 3 An arrow is launched ertically upward. It moes straight up to a maximum height, then falls to the ground. The trajectory of the arrow is noted. Which graph best represents the ertical elocity of the arrow as a function of time? Ignore air resistance; the only force acting is graity. 05 Pearson Education, Inc. Jun 5 0:5 AM Jun 5 0:5 AM Question 3 An arrow is launched ertically upward. It moes straight up to a maximum height, then falls to the ground. The trajectory of the arrow is noted. Which graph best represents the ertical elocity of the arrow as a function of time? Ignore air resistance; the only force acting is graity. Question 4: Analyzing a rock s fall A heay rock is dropped from rest at the top of a cliff and falls 00 m before hitting the ground. How long does the rock take to fall to the ground, and what is its elocity when it hits? Draw it.. yi = 00 m. D Jun 5 0:5 AM Jun 5 0:5 AM Question 4: Analyzing a rock s fall (cont.) Free fall is motion with constant acceleration: ay = g.. Using (y)i = 0 m/s and ti = 0 s, we find We can now sole for tf: Now that we know the fall time, we can use the first kinematic eq to find (y)f: 5 A ball is dropped from rest and falls (do not consider air resistance). Which is true about its motion? A acceleration is constant B elocity is constant C elocity is decreasing D acceleration is decreasing Jun 5 0:5 AM May 9 0:00 PM 3

33 6 A ball is thrown down off a bridge with a elocity of 5 m/s. What is its elocity seconds later? 7 An arrow is fired into the air and it reaches its highest point 3 seconds later. What was its elocity when it was fired? Sep 5 :38 PM Sep 5 :4 PM 8 A rocket is fired straight up from the ground. It returns to the ground 0 seconds later. What was its launch speed? Sep 5 :44 PM 33