Speed. Change of Distance (in meters) Change of Time (in seconds) T 5 sec. Step 2: Formula D. 100m in 10sec. 200m in 10sec T.

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1 Speed Unit 6:1 Speed Speed is how fast something is moving. Precisely, it is how far an object travels in a certain amount of time. The standard metric units are meters per second (m/s), but any units of distance divided by time will work (like miles per hour [mph] or cm per sec [cps], etc). Speed (in meter/sec) S = D Change of Distance (in meters) Change of Time (in seconds) Speed equal change of distance (distanced traveled) divided by change of time. Where D = D final D initial Ex. A plane flies meters in 5 sec. Calculate its speed. Step 1: Variables S = D = m = 5 sec Step : Formula D S = T Step 3: Put in numbers and solve S D = = 5 S = 4 Step 4: Check units S = 4 m/sec Why we use change of distance: A tree 4 m away for sec has a speed of zero it hasn t moved. That s why we have to use D (change of distance) instead of distance (D). An object has to be moving to have speed. Physics Explains Mathematics: If = (in S = D/), then an object is in two places at once, which is impossible. This is why dividing by zero is undefined: it makes no physical sense! Constant Speed If an object moves at constant speed, it travels the same amount of distance each second. Notice that there is equal space between each dot. Speed is proportional to distance: A faster object goes farther, in the same amount of time. 1m in 1sec m in 1sec Speed is indirectly proportional to time: A faster object travels the same distance in less time. m in sec D 1 S1 = = = 1m/s 1 Each dot represents an object s position at regular time intervals (time is constant). S Doubling the distance, doubles the speed. D = = = m/s 1 Doubling the time, halves the speed. D S1 = = = 1m/s D S = = = m/s m in 1sec 1 A slower object can travel the same distance as a faster object, it just takes more time. A fast object travels the same distance faster. Fast object Slow object Distance Traveled Measuring Speed Initial Position Final Position 5 m To measure speed you must measure the distance traveled and the elapsed time. Measure distance in meters using a meter stick or measuring tape. Measure time with a stopwatch or with photogates. Photogates (which start and stop when an object breaks beams of light) are a very accurate and precise method of measuring time. :. Elapsed Time 5 sec D 5 m S = = = 5 m /s T 5 sec :5. Copyright 8, C. Stephen Murray

2 Unit 6:1 1. Speed. Distance Traveled 3. Elapsed Time Constant Speed A. How far an object moves between two positions. B. When an object covers equal amounts of time each second. C. The rate of how fast an object travels a particular distance. D. How many seconds it takes for an event to occur. E. Delta: means change of. 1. Slow speed. Fast speed 3. Photogate 4. Directly Proportional 5. Indirectly Proportional A. An object that travels a long distance quickly. B. Can travel a long distance, but requires a lot of time. C. Uses a beam of light to start and stop a timer. D. One quantity increases as another quantity increases. E. One quantity decreases as another quantity increases. Will Speed Increase or Decrease? Mark these as Speed, Distance, Time, or Other Distance is constant and time increases. 5 mm/sec meters/sec 15 ft/min Time is constant and distance decreases. 1 inches 8 meters 78 sec Time is constant and distance increases. 5 m/s 8 minutes 6 Newtons Distance is constant and time decreases. True or false (and why): A fast car goes farther. start Can a slow object travel as far as a fast object? Explain. Why do we have to use change of distance ( D) instead of just distance (D)? 1. Is the above motion at constant speed?. Why or why not? 3. Each dot = 1 sec. How long did it take to go 15 m? 4. Calculate the object s speed. 5. How would the dots change if it were moving faster? A bike moves 5 m in 1 seconds. Calculate the speed of the bike. A car travels miles in 4 hours. Calculate the car s speed. Step 1: S = D = = Step 3: Plug in numbers and solve: Step 1: S = D = = Step 3: Plug in numbers and solve: Step : Formula: Step 4: Give answer with units: Step : Formula: Step 4: Give answer with units: A car travels 6 m/s for 1 secs. Calculate how far it traveled. On holiday, a family travels from Meyerville (1 miles away) to Sprytown (7 miles away), in 3 hours. Find their speed. Step 1: Step 3: Step 1: Step 3: Step : Step 4: Step : Step 4: Copyright 8, C. Stephen Murray

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4 Velocity and Acceleration Unit 6: Speed vs. Velocity Velocity is speed with direction. Example: A person walks 4 m/s speed (no direction). m/s north Same speed; different velocities because they have different directions. m/s west Velocity changes when direction changes. Scalars vs. Vectors Vectors require direction; Scalars only need magnitude (how big). Remember: Speed is a Scalar; Velocity is a Vector. Vectors require magnitude (how much) and direction, often vectors can cancel each other out (not acceleration, though). 1 m/s west Magnitude Direction Speed: 1 m/s. Velocity: 1 m/s west. Acceleration Acceleration is how fast you change velocity OR how much the velocity changed in a certain amount of time. An object accelerates when it changes speed OR changes direction! Acceleration (in m/s ) a = V Change of Velocity (in meters/sec) Change of Time (in seconds) Acceleration equal change of velocity divided by change of time. V V = V V, so, a = final initial final V initial Finding V. always = final initial. V = V final V initial OR Final velocity Initial velocity. If V is positive the object is speeding up. If V is negative the object is slowing down (see below). Ex. A plane starts at rest and ends up going m/s in 1 secs. Calculate its acceleration. Step 1: Variables V i = m/s (at rest) V f = m/s T = 1 sec a = Step : Formula V a = T Step 3: Put in numbers and solve V V f Vi a = = = 1 a = = 1 Step 4: Add units a = m/s Pos. means speeding up Ex. A race car starts at 4 m/s slows to 1 m/s in 5 seconds. Calculate the car s acceleration. Step 1: Variables V i = 4 m/s V f = 1 m/s T = 5 sec a = Step : Formula V a = T Step 3: Put in numbers and solve V V f Vi 1 4 a = = = 5 3 a = = 6 5 Step 4: Add units a = 6 m/s Neg. means slowing down Negative acceleration means an object is slowing down OR speeding up in the negative direction. Slowing down is also called deceleration. Distance and Acceleration An object that is accelerating will travel farther each second. Constant Speed Equal Distance Positive Acceleration Increasing Distance Points are equal distance, so velocity is constant. Since the velocity is constant, the initial and final velocity are equal and the acceleration equals zero. The distance between the points is increasing, so velocity is increasing. The object is accelerating: traveling faster each second and covering more distance every second. Measuring Acceleration To measure an object s acceleration you need to measure the object s velocity before and after the acceleration. If the object starts at rest you know that V i = m/s. If the object stops you know that V f = m/s. Measure V i (Initial Velocity) 4 m in 1 sec D 4 m V i = = T 1 sec V = 4 m /s in itia l Measure (Time it took to Accelerate) Accelerates for seconds So = sec V f Vi 8 4 a = = 4 Vinitial = = m/s Measure V f (Final Velocity) 8 m in 1 sec D 8 m V f = = T 1 se c V = 8 m /s fin a l Copyright 8, C. Stephen Murray

5 Unit 6: Speed (S) or Velocity (V) Scalar (S) or Vector (V) Mass, Time, Distance, Velocity, or Acceleration? A bike goes 5 m/s toward main street. A person walks 4 mph. 4 mph toward Dallas. 3 m/s to the left. 1 meters up the hill. hrs 3 m/s 6 mph/sec 5 sec 9 mph 1 m 8 kg 4 m/s 1 in A plane flies m/s. A bird flies 1 mph due south. 1 meter per sec. Direction matters. No direction is needed Object A Object B At constant velocity. Accelerating? Yes, No, or Maybe? Going 5 m/s then going 3 m/s. A car going around a corner. (see graphic at right). At constant speed. Stopping. A car at rest. 1 m/s 1 m/s Object A accelerates at 1 m/s ; Object B accelerates at 5 m/s. Which one will go faster? Which one will take more time to reach a high speed? If they start at rest, which one will reach 4 m/s first? Which one goes farther (longer distance)? Which one will be 1m away sooner? A person starts running from m/s to 6 m/s in seconds. Calculate the person s acceleration. Object C Object D Choose which of the above applies to the following Constant speed. Positive acceleration. At constant velocity. Accelerating. Decelerating. Acceleration =. Distance increases Starts at rest. Is stopping. Constant direction. Negative acceleration. V i = V f Give what you know for the following: (V i, V f, or a) An object at constant velocity. An object that is stopping. An object that accelerates from rest. An object at rest. A dragster s top acceleration is 6 m/s. If it starts from rest at the starting line, how fast will it be going after 3 seconds? Solve: Solve: Formula: Formula: A plane stops from 5 mph in 5 seconds. Calculate the planes acceleration. Solve: A car travels 3 m in 5 seconds. After accelerating for 3 seconds, it travels m in seconds. Calculate the car s acceleration. 1) Find V i. ) Find V f. Formula: 3) Calculate a. Copyright 8, C. Stephen Murray

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7 Graphing Linear Motion Unit 6:3 Position vs. Time Graphs A Position vs. Time graph shows where an object is at a particular time. The slope of a position vs. time graph shows the speed of an object. A steeper line shows faster speed. A downward line means negative speed (moving left or coming back). Position (m) Position vs. Time Line A fast speed slow speed Line B no speed 15 Line C 1 5 Line D Starting position (t = ) Time (sec) negative speed A steeper line = a faster speed. Object A travels 3 m in 5 seconds. Line A shows fast positive speed. Object B travels 3 m in 1 seconds. Line B shows slow positive speed. Object C stays 15 m away. Line C shows a speed of zero. Object D travels m in 1 seconds. Line D shows slow negative speed. S S S S LineA LineB LineC LineD D 3 = = = 6m/s 5 D 3 = = = 3m/s 1 D = = = m/s 1 D = = = m/s 1 Graphing Variables Scientists have rules for choosing which variable is graphed on which axis. This allows scientists to understand how an experiment was conducted just by reading the graph. Conventions: X-axis (horizontal): Independent or manipulated variable. Y-axis (vertical): Dependent or responsive variable. Independent vs. Dependent The independent variable is not affected by the changing dependent variable. The dependent variable changes as the independent variable Dependent variable Velocity (in m/s) Time (as in a particular moment in time ) is always an independent variable (x-axis) because nothing stops time. Time does not change with speed; speed changes over time. Velocity vs. Time Time (in sec) Independent variable Manipulated vs. Responsive Sometimes it is hard to determine which is the independent variable. In these cases, the variable that you are manipulating (varying) will graphed on the x-axis. Duration (how long it takes) can be dependent (y-axis). Ex. The period of a spring (how long it takes to move back and forth) changes as more mass is added. Mass is independent, not period of time. Responsive variable Acceleration (in m/s ) Acceleration vs. Force Force (in N) Manipulated variable The above object s acceleration changes (responds) as the force is changed (manipulated). The manipulated variable is the one you are changing in your experiment and is often the experimental variable. Meaning of Slope Changes The slope of a position vs. time graph is speed. The slope of a velocity vs. time graph is acceleration. Yet for some graph, the slope has no physical meaning. To figure out what the slope of a graph means: divide the y-axis units by the x-axis units to find the units for the slope. rise Meaning of Slope = run units of y-axis = units of x-axis Velocity (in m/s) Velocity vs. Time Time (in sec) This graph shows the change of velocity over time which is acceleration. Slope rise run Slope = acceleration y x m/s s = = = = m/s = acceleration Acceleration (in m/s ) Acceleration vs. Time Time (in sec) This graph shows the change of acceleration over time which is undefined. rise y m/s Slope = = = = = run x s The slope of this graph means nothing. 3 m/s? Copyright 8, C. Stephen Murray

8 1. Linear. Responsive variable 3. Independent variable 4. Dependent variable 5. Slope 6. Manipulated variable Position (m) Position vs. Time A A. Vertical axis (y) variable. B. The variable you change. C. Any straight line graph. D. Measure of how steep a line is. E. The variable on the horizontal axis (x-axis). F. What changes because you change something Time (sec) B D C Circle the Independent Variable A. Time or Acceleration B. Velocity or Time C. Time or Position Circle the Manipulated Variable for these Graphs A. Force on an object or Acceleration of the object? B. Period of a Spring or Mass hung from the spring? C. Number of batteries or Brightness of a bulb? What does the slope of this line show? How much time does it take Object A to travel 1m? How much time does it take Object B to travel 1m? Which Object (A or B) has the faster velocity? Object C starts where? Object C ends where? Which line shows negative speed? Which line shows positive speed? Which line shows an object at rest? What is Object D s initial position? Unit 6:3 Velocity (m/s) Velocity vs. Time Time (secs) When was the object moving at 15 m/s? How fast is the object going after 1 seconds? What was the initial velocity of the object? How much speed does it gain in the first 5 seconds? Find the slope of the graph (must show work) What does the slope you just found stand for? Position (m) Position vs. Time Time (sec) Which is the independent variable? Which is the dependent variable? Where was the object at 4 seconds? Where did the object begin? Find the slope of the graph (must show work) What does the slope you just found stand for? The slope of this graph means: Which segment shows: Increasing velocity: Constant velocity: Positive acceleration: Negative acceleration: Speeding up: Slowing down: Velocity A Velocity vs. Time B C Time D Position Position vs. Time Time A B C D Which segments shows: At rest: Fast speed: Slow speed: Going backwards: Going forward: Negative speed: Speed equals zero: Copyright 8, C. Stephen Murray

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10 The Linear Equation y - axis variable Must be for the same point y = mx + b slope x - axis variable y-intercept The linear equation is the form of ANY straight line. The linear equation is just a formula and like any other formula you can solve for any unknown given the other variables. For example, if you are given x and y for a point and the y-intercept (b), you could solve for the slope of the line. Step 1: Calculate slope (m) with two good points (where the line hits cross hairs [see graph]). rise y 4 m/s m = = = = 4 m/s run x y = 1 s 8 4 = 4 m/s The slope (the tilt) tells you the rate of change of y, not x. In this case slope tells us the change of velocity which is acceleration (notice the units: m/s ). More slope (more tilt) would mean more acceleration, the velocity would change faster. Step : Find the y-intercept (b) (where the line crosses the y-axis ). b = m/s Velocity (m/s) y intercept Velocity vs. Time Time (sec) x = = 1 s Another good point The y-intercept (the vertical shift) tells you the initial condition of the object: this object s initial velocity = m/s (velocity at sec). Step 3: Find what the x and y variables are for this graph. y-axis = velocity = v (in m/s) x-axis = time = t (in sec) Why is this step so important? If you leave x and y in the linear equation it is easy to be confused when putting in numbers. Which one is time? Which one is velocity? If you change your variables there will be no confusion. The y variable for this graph is velocity, v. Velocity (m/s) Velocity vs. Time Time (sec) The x variable for this graph is time, t. x and y will be different for each graph! Step 4: Put all of the above into the linear equation to find the equation for this particular line. Example: When will the object graphed above be going m/s? y = v x = t m = 4 m/s b = m/s y = mx + b v = 4t + Any line THIS LINE Solution: use the linear equation for this line. v = m/s t = v = 4t + = 4t + = 4t 18 = 4t t = 18/4 t = 4.5 sec Step 5: Use the linear equation to solve problems. You now have a formula for the object depicted on the graph. Given any x or y you can now solve for the other. The object will be going m/s at 4.5 seconds. (Notice this is a point beyond the graph. This is known as extrapolation. Extra = outside.) Copyright 1, C. Stephen Murray

11 1. m, b, x, or y? A. vertical axis. B. Slope C. y-intercept D. horizontal axis E. Dependent variable.. Write the equation for slope. 3. Write the equation that defines a line. F. Gives initial condition. G. Independent variable H. Rate of change of y. I. Are constants. J. Are variables. 5. Use the two graphs below to answer the following questions. A. What is the y variable for Graph B? B. What is the x variable for Graph A? C. What is the y-intercept for Graph A? D. What is the y-intercept for Graph B? Position (in m) Time (in sec) Velocity (in m/s) Time (in sec) Acceleration (in m/s ) A B C 4. Use the graphs above to answer the following. A. What is the y variable for graph C? B. What is the x variable for graph B? C. What is y for graph A? D. What is x for graph B? E. In the linear equation what is y for graph B? E. Over time, what changes in Graph A? F. So, what does the slope of Graph A show? G. Over time what changes in Graph B? H. So, what does the slope of Graph B show? Linear Equation p Time (in sec) Graph A 1 Position vs. Time Graph B 8 Velocity vs. Time 1 7 Position (m) Time (sec) Velocity (m/s) Time (sec) 6. Use Graph A above to answer the following questions. 7. Use Graph B above to answer the following questions. A. On the above graph, calculate the line s slope. B. Put a square around the y-intercept. C. Write the linear equation variables for this line: A. On the above graph, calculate the line s slope. B. Put a square around the y-intercept. C. Write the linear equation variables for this line: m = b = y = x = D. Write the linear equation for this line: m = b = y = x = D. Write the linear equation for this line: E. Meters would go into what part of this linear equation? E. Seconds would go into what part of this linear equation? F. At what time will the object be at 15 meters? F. How fast is the object going after 1.5 seconds? G. What is the initial position of the object? G. What is the initial velocity of the object? Copyright 1, C. Stephen Murray

12 1 in 1ft Conversions are how we change units. 1 foot equals 1 inches: the amount is the same, but how we express the amount is different. Conversions Conversion Factors To do a conversion you need a conversion factor. Conversion factors come from equalities. Since anything divided by itself is 1, a conversion factor also equals 1. Any equality can make two conversion factors. 1 Equality 1 in = 1 ft Conversion Factors 1 in 1 ft OR 1 ft 1 in How To Do Conversions Follow these steps exactly and you will be able to perform any conversion. Example 1: Convert 35 m/s to ft/sec. Step 1: Write what you are given as a fraction with one unit on top and one unit on bottom. 35 m/s (given) becomes 35 m ft 1 sec m 35 m 3.3 ft 1 sec 1 m 35 m 1 sec Step : In parenthesis and WITHOUT NUMBERS, write the units you want to get rid of diagonal from itself. In the other part of the fraction write what you re converting to. NO NUMBERS YET! Notice: m s are diagonal. Step 3: Put numbers into the parenthesis so that the top equals the bottom. Since we know 3.3 ft = 1 m. Step 4: Cancel out the units BUT NOT THE NUMBERS! 35 m 3.3 ft 1 sec 1 m m s cancel because m/m = 1 Step 5: Do the math. Multiply the numbers if they are both on top. Divide if the second one is on the bottom. 35 m 3.3 ft 35(3.3) ft = = ft/sec 1 sec 1 m 1 sec If you have a single unit, just write it over ft becomes 15 ft 1 Why no numbers? Because most mistakes are made by assuming that you will multiply or divide by some number. Let the units guide you NOT the numbers. Ex. : Convert 5 mi/hr to mi/min. 5 mi Step 1: write as a fraction 1 hr 5 mi hr hr s are diagonal Step : 1 hr min from each other 5 mi 1 hr Step 3: put in #s 1 hr 6 min 5 mi 1 hr Step 4: since hr/hr = 1 1 hr 6 min 5 mi 1 hr 5 mi 6 on Step 5: = bottom 1 hr 6 min 6 min means = 5 / 6 =.83 mi/min Multiple Conversions Convert: 56 hours to weeks. Chaining: 56 hr 1 days 1 w eeks 1 4 hr 7 days 56 w eeks = = 3.33 w eeks 4(7) If you need to perform multiple conversions, you can either do each conversion independently or in one long chain. Convert: 56 hours to weeks. One conversion at a time: 56 hr 1 days = 3.33days 1 4 hr 3.33 days 1 weeks = 3.33 weeks 1 7 days Both ways will give the same answer, but once you master the chaining method, you will find it easier and less prone to mistakes. Copyright 1, C. Stephen Murray

13 Conversions p 1. Prepare these numbers for conversion. A. Ex. 1 in B. 6 m/sec C. 4 sec D. 19 mph E. 3.7 meters 1 in 1. Find the mistakes in each of the following and write a corrected version underneath. A. 4 mph 5,8 ft B. 5. m 1 min C. 8 years 3 days D. 1 1 mi 1 sec 6 sec 1 1 year 4 in 1 ft 1 in 3. Perform the following functions (do the math). A. B. C. 6 4 km km = = D. E. F. m sec 16 m 1 m = = sec min 1 sec 3.3 ft km 1 = 1 km sec 1 min = 1 6 sec 4. Do the following conversions. Given: 1 in =.54 cm; 3.3 ft = 1 m; 1 in = 1 ft; 5,8 ft =1 mi (mile) 5. Convert 1 m/min to m/hour. A. Convert 3.5 miles to feet 6. There are 1,, micrometers (µm) per meter. How many meters is 48, µm? B. Convert 6 ft to meters 7. A. Convert 15 in/min to feet per min C. Convert.5 weeks to days B. Using the above answer, convert to feet per second. D. Convert 5 seconds to minutes 8. A. Convert 54 cm/min to cm/sec E. Convert 18 m/sec to m/min B. Convert to inches per second. F. Convert 6 mph (miles) to m/hr (meters) 9. Convert 1 mph (miles) to m/s (meters). Copyright 1, C. Stephen Murray

14 Position, Distance, Displacement Position, distance, and displacement are all measured in meters, but they have different physical meanings. Position (x) Position is where you are relative to a reference point. x i is the initial position. x f is the final position. Distance (D) Distance is how far you have traveled between two positions. Distance is always positive. Displacement ( x) Displacement is the straight line distance between the initial and final positions. x = x f x i. Displacement can be positive or negative. In this example the displacement and distance are the same amount, but the displacement is negative, because they moved to the left. Final position: x f = 4 m Distance: D = 7 meters Displacement: x = 7 m = -4 3 Initial position: x i = 3 m But what if an object turns around? The distance traveled would continue to increase, but the displacement would begin to decrease as the final position became closer to the initial position. If it were to return to its initial position, its displacement would be zero. Total Distance: D = 9 meters Displacement: x = 3 meters D 1 = 6 m Reference point x D = 3 m in meters in meters in meters in meters An object that travels a circular path and ends up at its starting point has a distance equal to the circumference of the circle: D = πr. Yet the displacement is zero because it ended up where it started: its initial and final positions are the same. D = πr r x = m x f x i Remember that displacement is the straight line distance between the initial and final positions. In some cases you may need to use Pythagorean theorem to find x: A + B = C. x = 5 m (since = 5 ) D 1 = 4 m D = 3 m x f x i Vertical Displacement ( y) y is just like x except it is up or down.. + y means the final position is above the initial. y means the final position is below the initial. x and y When an object moves at an angle we can find both the x and y displacements independently. If an object moves up or down we use y, not x. Remember that down is negative, so a falling object will have a negative displacement. y Initial position initial + x y final In this example, the object has a positive x-displacement (because it moved to the right) and a negative y-displacement (because it fell). Copyright 1, C. Stephen Murray

15 1. Use the number line at the right to answer the following questions. A. What is the position of letter A? x A = B. What is the position of letter C? x C = C. What is the distance from A to C? D. What is the distance from D to A? E. What is the displacement from D to A? -5 A B C D in meters in meters I 4 m 3 m II. A. If II is the reference point, what is the position of the car at I? B. What is the total distance the car traveled? D = C. What is the car s first displacement from I to II? III D. What is the total displacement of the car from I to III: x = 3. A. What is the curved distance from a to c? B. What is x from a to c? C. What is the curved distance from c to a? D. What is x from c to a? E. What is the distance 1 time around the circle? F. What is the displacement 1 time around? 4. A ball is thrown horizontally from the top of a 7 m tall ledge. A. What is its vertical displacement during the fall? y = B. What is its horizontal displacement? x = C. What is the total displacement (straight line) from start to finish? 5. The grid at the right is 1 m between each of the horizontal and vertical rows. A. From D to E: x = y = B. From A to M: x = y = C. From B to O: x = y = D. Draw this path: D to B to J to L: i. x = ii. y = iii. D total = E. What is the total displacement (straight line) from B to P? Copyright 1, C. Stephen Murray

16 Kinematic Equations (R) Displacement ( x) in m Distance (D) is how far an object has traveled. Displacement ( x) is how far an object has moved from its original position. Displacement can be positive or negative. D = πr r start x = m stop An object that travels a circular path and ends up at its starting point has a distance equal to the circumference (πr), but no displacement. Vertical Displacement If an object moves up or down we use y, not x. Remember that down is negative (as is moving to the left for x). y Velocity (v) in m/s Velocity is how fast an object changes position. V i = initial velocity; V f is final velocity. When an object turns around v = m/s. V is if moving to the left V is + if moving to the right V is + when an object moves up. V is when an object moves down. Acceleration (a) in m/s A positive acceleration occurs when an object speeds up in the positive direction or slows down in the negative direction. Acceleration is how fast an object changes velocity. The kinematic equations work only with constant acceleration. Acceleration can be positive, negative, or zero. V f is V f is a is + V i is A negative acceleration occurs when an object speeds up in the negative direction or slows down in the positive direction. a is V i is + Time (t) in sec Time is always elapsed time, not a point in time. Also, time in any other units other than seconds must be converted first. Kinematic Equations With these five equations you are able to calculate for any unknown in linear motion. 1 x = ( vi + v f ) t v f vi a = t 1 x = ( vit) + a( t) 1 x = ( v f t) a( t) v = v + ( a x) f i Choosing an Equation a is not used x is not used v f is not used v i is not used t is not used Just as with any other word problem, first write a variable list from the given information, including your unknown. Then choose an equation which includes these variables. Big Trick: Figure out which variable is not used in your variable list, then chose the equation that is also not using this variable. Remember that your unknown is still in your list, you just don t know its value yet. If your unknown is not in the equation, you can t solve for it. x = 5 m t = 1 sec a = m/s Vf = Vi is not used in our list. Vf is on this list: it is the unknown. So choose this equation because it does not use Vi and has all of your variables. 1 x = ( v f t) a( t) Example 1: An object moves 1 m to the left in 4 seconds. If its initial velocity was 5 m/s to the right, what is the acceleration of the object? x = 1 m (moves left) t = 4 sec Vi = 5 m/s (right is +) a = V f is not in this list Vf is not used = + a(16) 1 = 5(4) + a(4) 1 x = ( vit ) + a ( t ) 1 = + (8 a) 3 = 8a a = 4m/s Example : An object at rest ends up moving m/s to the right after traveling 8 meters to the right. How much time did this take? Vi = m/s (at rest) Vf = m/s x = 8 m t = a is not in this list. a is not used 1 x = ( vi + v f ) t 1 8 = ( + ) t 1 8 = () t 8 = 1t t = 8 sec Copyright 1, C. Stephen Murray

17 1. x, y, t, v i, v f, or a? Kinematic Equations p 6. Choose the correct kinematic equation for the following: sec 3 m/s 6 m right How far... 4 m/s How fast.... A person swims to the other end of a m long pool and back. What is their displacement? 3. A rock falls 15 m. Is this vertical or horizontal motion? What is the displacement of the rock? 4. A car moving 1 m/s stops in 3 seconds. V f = How long did it take? How high You throw a rock into the air and catch it as it returns. What is the displacement of the rock? a = m/s V i = 6 m/s V f = 6 m/s x = a = 4 m/s t = 1 s V f = m/s x = a = 3m/s V i = 6 m/s V f = 1 m/s t = What s missing from the list: So use this equation: What s missing from the list: So use this equation: What s missing from the list: So use this equation: 7. In 1 seconds a car accelerates 4m/s to 6 m/s. How fast was the car going before it accelerated? 8. A object moving m/s experiences an acceleration of 3m/s for 8 seconds. How far did it move in that time? Equation and Solve: Equation and Solve: 9. An object at rest starts accelerating. If it travels 4 meters to end up going m/s, what was its acceleration? 1. A model rocket climbs m in 4 seconds. If was moving 1 m/s to begin with, what is its final velocity? Equation and Solve: Equation and Solve: 11. A car stops in 1 m. If it has an acceleration of 5m/s, how long did it take to stop? 1. An object drops m from a cliff. If it started at rest and is going m/s just before it hits the ground, what is its acceleration? Copyright 1, C. Stephen Murray

18 Free-fall is the expression we use for any object falling in our earth s gravitational field with only gravity acting on it: it is falling freely. On the earth the acceleration due to gravity ( g ) is 9.8 m/s. Because we usually call up the positive direction, g is given a negative value. For objects in freefall: a = g = 9.8 m/s Freefall Not all falling objects are in freefall. Parachutes, balloons, and airplanes all have air resistance or buoyancy slowing them down: a 9.8 m/s. Without air resistance light and heavy objects fall at the same rate. This can be proven in a vacuum chamber when all of the air is removed. On the moon, Apollo 15 astronauts showed this by dropping a feather and a hammer at the same time. They hit the ground at the same time. The moon has no atmosphere so it is a vacuum. It has gravity, but no air resistance. Special Situations Because a = g, very little information is needed to be able to solve a freefall problem, but often you must use your everyday knowledge to pull additional information out of a problem. Dropped objects: y is ; v i = m/s. Returns to initial position: y =, and v f = v i. y v i = m/s Dropped objects begin at rest and go down, so y is and v i = m/s. Examples: is dropped ; pushed off a ledge ; sitting on a cliff. v i = 5 m/s v f = v i = 5 m/s y = If an object comes back to its starting position then y = m and v f = v i. Examples: Back to the ground ; back to your hand ; from ground to ground. Final position at top: V f = m/s. v f = m/s v i = 5 m/s + y If the object s final position is at the top, then y is + and v f = m/s. Examples: How high does it go? ; find maximum height. The kinematic equations become the vertical kinematic equations just by putting y in for x. Choose the correct equation by deciding which variable is not used in your problem. Vertical Kinematic Equations 1 y = ( vi + v f ) t v = v + ( at) f y = ( vit) + a( t) y = ( v f t) a( t) f i i 1 1 v = v + ( a y) a is not used y is not used v f is not used v i is not used t is not used Example 1. An object is dropped from 4 m. How fast is it going at the bottom? Dropped so: v i = m/s Falling so: y = 4 m a = 9.8 m/s v f = (t is not used) f v = v + a y f i v = +(-9.8)(-4) f v = 784 f v = 784 = ± 8 Because it is going down we choose the negative: v f = 8 m/s Example. An object is thrown up into the air going 8 m/s. How long does it take for it to get back to the ground? v i = 8 m/s Because it comes back to its original position: y = m v f = 8 m/s a = 9.8 m/s t = Because we have all of the variables, we choose the easiest equation. v = v + at f -8 = 8 + (- 9.8t) t = -16 = -9.8t i sec -9.8 = Notice that mass is not in the equation, meaning two objects of different mass will hit the ground at the same time! Copyright 1, C. Stephen Murray

19 1. Fill in the missing information. V B = a = C B D V C = a = V D = 4m/s a =. Freefall? Yes or No? An airplane. A volleyball hit over a net. 3. What do we call any space that has no air? 4. If the two objects at the right are dropped in a vacuum, which will hit the ground first? Paper floating down. A ball rolling off a table. A person jumping. kg Freefall p 5 kg 5. What if there is air resistance? V A = 1m/s a = A E The Ground V E = a = 6. An object is thrown 3 m/s from the ground and it lands on the ground. v i = ; v f = ; a = ; y =. 7. An object is thrown into the air going 8 m/s. How high does it go? v i = ; v f = ; a = ; 8. An object is dropped from a 15 m ledge. How fast it is moving just before it hits the ground? Equation and Solve: 9. A person throws tennis ball 6 m/s straight up. How long does it take for it to come back to their hand? Equation and Solve: 1. A ball is thrown 4 m/s into the air. How high does it go? 11. A rock falls off a cliff and falls for 3 secs. How high was Equation and Solve: the cliff? Equation and Solve: 1. An object is thrown up into the air going 9 m/s. How fast is it going seconds later? 13. An object is thrown 16 m/s straight up from a 7 m tall cliff. How much time does it take to hit the ground below? Copyright 1, C. Stephen Murray

20 Linear Motion In Class Test Review This is NOT the homework!!! 1) Circle the bigger one: A. Centimeters or megameters? B. Micrometers or millimeters? C. Kilometers or megameters? D. Centimeters or millimeters? ) Convert 18 m/s to meters per min. 3) An object moves 1 m in 15 seconds. Calculate the object s speed. 4) An object moves 18 m/s. How long does it take the object to move 154 m? 5) Speed or velocity: A person walks.5 m/s to the east. 6) Scalar or vector: A car is moving 3 m/s. Object A Object B Object C Object D 7) The tape timers at the left show 4 objects moving to the right. The dots show the positions of the objects each second. Which objects apply to the following? Constant speed. Positive acceleration. At constant velocity. Accelerating. Decelerating. Acceleration =. Distance increases Starts at rest. Is stopping. Constant direction. Negative acceleration. V i = V f 8) A car begins at a stop sign. It ends up going 1 m in 6.5 seconds. Find the car s acceleration. Equation and solve: 9) +,, or? A. Acceleration of an object that is moving to the left and speeding up? B. Acceleration of an object that is moving up and slowing down? C. Velocity of an object that is moving to the right? D. Displacement of an object that ends at its starting position? E. Acceleration of an thrown object at the top of its path? F. Displacement of an object moving to the left? 1) What is the acceleration of a full bottle of water dropped from a desk? An empty bottle? 11) When an object is dropped or thrown into the air, what is its acceleration? 1) An object dropped from a 4 m tall roof. y = and v i =. 13) An object is thrown 1 m/s into the air. How high does it go? v i = ; a = ; and v f =. 14) A person throws a ball into the air at 6 m/s from the ground. When it comes back, v f =, a =, and y =. 15) Sitting on the dock of the bay, wasting time with my sister. I get bored and push her off the m dock. How fast is she moving when she belly flops into the water? (And more importantly how badly is she going to hurt me when she catches me?) Equation: Solve: 16) What is the velocity of the stop sign in the car s frame of reference? 17) What is the motorcyclist s velocity relative to the car? 3 m/s m/s Copyright 1, C. Stephen Murray

21 :. Constant :4. speed Accelerating In Class Review p :7. Constant :9. speed 1 m 4 m 18) In the graphic above, the car is at constant speed between the first two positions and between the last two positions. Between the middle two positions it is accelerating. Calculate its acceleration. Use the three motion graphs below to answer the following questions. 19) What does the slope of the graphs below tell us: Graph 1: ; Graph : ; Graph 3:. ) Transfer the following graphs. Each vertical square is 1 m; each horizontal square is 1 sec. Position vs. Time Velocity vs. Time Acceleration vs. Time Position A B C Velocity Acceleration Time Time 1) Use the graph at the right to answer the following. A. Give the linear equation for the graph at the right. Velocity (m/s) B. Where is the object on the graph at 4. seconds? C. What does the y-intercept tell us about this object? D. What is the speed of the graph? E. Transfer the position graph to the velocity and acceleration graphs below Velocity vs. Time Time (sec) Position (m) Acceleration (m/s) Time Position vs. Time Time (sec) Acceleration vs. Time Time (sec) Copyright 1, C. Stephen Murray

22 Instantaneous Vs. Average Speed Instantaneous Speed Velocity at a particular instant. How fast something is moving at a particular point in time. This is what your speedometer reads. Average Speed The average velocity over an entire distance. Average velocity is found from total distance divided by the total time. Average Speed (in meter/sec) v ave D = t total total Total Distance (in meters) Total Time (in seconds) Fuel Stop 3 min 6 mi.75 hr Banalville Home 8 mi 1.3 hr Gastin 11 mi 1.5 hr Sightseeing 3 min Porkerville Pulchritude 5 miles 1.4 hr Stop at Fatties Finest Foods 1. hr 75 miles 1 hr Destiny The diagram shows a person s circuitous journey. During any trip your speed does not stay constant due to different speed limits, traffic, stops, etc. To find the average speed between any two points, you need total distance and total time between those two points. Instantaneous Vs. Average Speed Instantaneous Speed Velocity at a particular instant. How fast something is moving at a particular point in time. This is what your speedometer reads. Average Speed The average velocity over an entire distance. Average velocity is found from total distance divided by the total time. Average Speed (in meter/sec) v ave D = t total total Total Distance (in meters) Total Time (in seconds) Fuel Stop 3 min 6 mi.75 hr Banalville Home 8 mi 1.3 hr Gastin 11 mi 1.5 hr Sightseeing 3 min Porkerville Pulchritude 5 miles 1.4 hr Stop at Fatties Finest Foods 1. hr 75 miles 1 hr Destiny The diagram shows a person s circuitous journey. During any trip your speed does not stay constant due to different speed limits, traffic, stops, etc. To find the average speed between any two points, you need total distance and total time between those two points. Copyright 1, C. Stephen Murray

23 Linear Motion In Class Test Review 1) Convert 6 ft/s to m/s ) Convert 18 m/s to meters per min. 3) An object moves 1 m in 15 seconds. Calculate the object s speed. 4) An object moves 18 m/s. How long does it take the object to move 154 m? 5) Speed or velocity: A person walks.5 m/s to the east. 6) Scalar or vector: A car is moving 3 m/s. Object A Object B Object C Object D 7) The tape timers at the left show 4 objects moving to the right. The dots show the positions of the objects each second. Which objects apply to the following? Constant speed. Positive acceleration. At constant velocity. Accelerating. Decelerating. Acceleration =. Distance increases Starts at rest. Is stopping. Constant direction. Negative acceleration. V i = V f 8) A car begins at a stop sign. It ends up going 1 m in 6.5 seconds. Find the car s acceleration. Equation and solve: 9) +,, or? A. Acceleration of an object that is moving to the left and speeding up? B. Acceleration of an object that is moving up and slowing down? C. Velocity of an object that is moving to the right? D. Displacement of an object that ends at its starting position? E. Acceleration of an thrown object at the top of its path? F. Displacement of an object moving to the left? 1) What is the acceleration of a full bottle of water dropped from a desk? + or - 11) When an object is dropped or thrown into the air, what is its acceleration? + or - 1) An object moves from rest to 4 m away. x = and v i =. 13) What are two ways a velocity can change? 14) What does the slope of this velocity vs. time graph mean? 15) A shopping cart is going 4. m/s. It undergoes -5. m/s of acceleration for 4 seconds. How fast is it going afterwards? Equation: Solve: 16) If our velocity is positive and our acceleration is negative what is happening? 17) An object at rest accelerates for 6 seconds. Afterwards it is going 6 m/s. How far it traveled in this time? Equation: Solve:

24 :. Constant :4. speed Accelerating In Class Review p :7. Constant :9. speed 1 m 4 m 18) In the graphic above, the car is at constant speed between the first two positions and between the last two positions. Between the middle two positions it is accelerating. Calculate its acceleration. Use the three motion graphs below to answer the following questions. 19) What does the slope of the graphs below tell us: Graph 1: ; Graph : ; Graph 3:. ) In Graph 1, which letter has the highest velocity. Position vs. Time Velocity vs. Time Acceleration vs. Time Position A B C Velocity Acceleration Time Time 1) Use the graph at the right to answer the following. A. Give the linear equation for the graph at the right. B. Where is the object on the graph at 4. seconds? C. What does the y-intercept tell us about this object? D. What is the speed of the graph? 5 m/s Position (m) Time Position vs. Time Time (sec) 1 m/s. How fast does the big car seem to be moving to a person looking from the little car (in the car s frame of reference)? 3. A race car is going at 3 m/s but crashes into a wall. The crash lasts 4 seconds. Calculate the car's deceleration. *Remember if an object is slowing down or stoping what the acceleration will be. Equation: Solve:

25 Copyright 1, C. Stephen Murray

26 Copyright 1, C. Stephen Murray