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!T 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. plane flies 00 meters in 5 sec. Calculate its speed. Step 1: Variables S =!D = 00 m!t = 5 sec Step : Formula D S = T Step 3: Put in numbers and solve S 00 = = 5 S = 40 Step 4: Check units S = 40 m/sec Why we use change of distance: 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). n object has to be moving to have speed. Physics Explains Mathematics: If!T = 0 (in S =!D/!T), 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: faster object goes farther, in the same amount of time. 0m in sec 00m in sec Speed is indirectly proportional to time: faster object travels the same distance in less time. 00m in 0sec 0 S1 = = = m/s Each dot represents an object s position at regular time intervals (time is constant). S Doubling the distance, doubles the speed. 00 = = = 0m/s Doubling the time, halves the speed. 00 S1 = = = m/s 0 00 S = = = 0m/s 00m in sec slower object can travel the same distance as a faster object, it just takes more time. 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. 0:00.0 Elapsed Time 5 sec D 5m S = = = 5m/s T 5sec 0:05.0
Unit 6:1 1. Speed. Distance Traveled 3. Elapsed Time 4.! 5. Constant Speed. 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. n 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 0 meters/sec 15 ft/min Time is constant and distance decreases. inches 8 meters 78 sec Time is constant and distance increases. 50 m/s 8 minutes 6 Newtons Distance is constant and time decreases. True or false (and why): 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? bike moves 50 m in seconds. Calculate the speed of the bike. car travels 00 miles in 4 hours. Calculate the car s speed. Step 1: Variables: S =!D =!T = Step 3: Plug in numbers and solve: Step 1: Variables: S =!D =!T = Step 3: Plug in numbers and solve: Step : Formula: Step 4: Give answer with units: Step : Formula: Step 4: Give answer with units: car travels 60 m/s for secs. Calculate how far it traveled. On holiday, a family travels from Meyerville ( miles away) to Sprytown (70 miles away), in 3 hours. Find their speed. Step 1: Step 3: Step 1: Step 3: Step : Step 4: Step : Step 4:
Velocity and cceleration Unit 6: Speed vs. Velocity Velocity is speed with direction. Example: person walks 4 m/s speed (no direction). 0 m/s north Same speed; different velocities because they have different directions. 0 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. cceleration cceleration is how fast you change velocity OR how much the velocity changed in a certain amount of time. n object accelerates when it changes speed OR changes direction! cceleration (in m/s ) a =!V!T Change of Velocity (in meters/sec) Change of Time (in seconds) cceleration 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. plane starts at rest and ends up going 00 m/s in secs. Calculate its acceleration. Step 1: Variables V i = 0 m/s (at rest) V f = 00 m/s T = sec a = Step : Formula V a = T Step 3: Put in numbers and solve V V f Vi 00 0 a = = = 00 a = = 0 Step 4: dd units a = 0 m/s Pos. means speeding up Ex. race car starts at 40 m/s slows to m/s in 5 seconds. Calculate the car s acceleration. Step 1: Variables V i = 40 m/s V f = m/s T = 5 sec a = Step : Formula V a = T Step 3: Put in numbers and solve V V f Vi 40 a = = = 5 30 a = = 6 5 Step 4: dd 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 cceleration n object that is accelerating will travel farther each second. Constant Speed Equal Distance Positive cceleration 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 cceleration 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 = 0m/s. If the object stops you know that V f = 0m/s. Measure V i (Initial Velocity) 4 m in 1 sec D 4m V i = = T 1sec V = 4m/s initial Measure!T (Time it took to ccelerate) ccelerates for seconds So!T = sec V f Vi 8 4 a = = 4 Vinitial = = m/s Measure V f (Final Velocity) 8 m in 1 sec D 8m V f = = T 1sec V = 8m/s final
Unit 6: Speed (S) or Velocity (V) Scalar (S) or Vector (V) Mass, Time, Distance, Velocity, or cceleration? bike goes 5 m/s toward main street. person walks 4 mph. 40 mph toward Dallas. 3 m/s to the left. meters up the hill. hrs 3 m/s 6 mph/sec 5 sec 9 mph 1 m 8 kg 4 m/s 1 in plane flies 00 m/s. bird flies 0 mph due south. 1 meter per sec. Direction matters. No direction is needed Object Object B t constant velocity. ccelerating? Yes, No, or Maybe? Going 5 m/s then going 3 m/s. car going around a corner. (see graphic at right). t constant speed. Stopping. car at rest. m/s m/s Object accelerates at 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 40 m/s first? Which one goes farther (longer distance)? Which one will be 0m away sooner? 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. t constant velocity. ccelerating. Decelerating. cceleration = 0. 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) n object at constant velocity. n object that is stopping. n object that accelerates from rest. n object at rest. dragster s top acceleration is 60 m/s. If it starts from rest at the starting line, how fast will it be going after 3 seconds? Variables: Solve: Variables: Solve: Formula: Formula: Variables: plane stops from 50 mph in 5 seconds. Calculate the planes acceleration. Solve: car travels 30 m in 5 seconds. fter accelerating for 3 seconds, it travels 0 m in seconds. Calculate the car s acceleration. 1) Find V i. ) Find V f. Formula: 3) Calculate a.
Graphing Linear Motion Unit 6:3 Position vs. Time Graphs 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. steeper line shows faster speed. downward line means negative speed (moving left or coming back). Position (m) 35 30 5 Position vs. Tim e Line fast speed slow speed Line B 0 no speed 15 Line C 5 0 Line D 0 1 3 4 5 6 7 8 9 11 1 Starting position (t = 0) Time (sec) negative speed steeper line = a faster speed. Object travels 30 m in 5 seconds. Line shows fast positive speed. Object B travels 30 m in seconds. Line B shows slow positive speed. Object C stays 15 m away. Line C shows a speed of zero. Object D travels 0 m in seconds. Line D shows slow negative speed. S S S S Line LineB LineC LineD 30 = = = 6m/s 5 30 = = = 3m/s 0 = = = 0m/s 0 = = = m/s 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 cceleration (in m/s ) cceleration 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 cceleration (in m/s ) cceleration 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?
1. Linear. Responsive variable 3. Independent variable 4. Dependent variable 5. Slope 6. Manipulated variable Position (m) 10 0 80 60 40 0 0 Position vs. Time. Vertical axis (y) variable. B. The variable you change. C. ny 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. 0 4 6 8 1 Time (sec) B D C Circle the Independent Variable. Time or cceleration B. Velocity or Time C. Time or Position Circle the Manipulated Variable for these Graphs. Force on an object or cceleration 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 to travel 0m? How much time does it take Object B to travel 0m? Which Object ( 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) 350 300 50 00 150 0 50 0 Velocity vs. Time 0 1 3 4 5 6 7 8 9 11 Time (secs) When was the object moving at 150 m/s? How fast is the object going after 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 18 16 14 1 8 6 4 0 0 1 3 4 5 6 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 Velocity vs. Time B C Time D Position Position vs. Time Time B C D Which segments shows: t rest: Fast speed: Slow speed: Going backwards: Going forward: Negative speed: Speed equals zero:
Linear Motion Review Unit 6:5 mv = m times v F/a = F a T + T 1 = T T 1 mv = m v!d/!t =!D!T Match the variables with the quantities. 1. a = sec. S or v = m/sec 3. D = 43 m/s 4. F = 45 meters 5. T = newtons Equation: S =!D/!T; solve for!d. If!v = v v 1, solve for v : If p = mv, solve for m. If a =!V/!T, solve for!t: What do you need to know in order to find an object s speed? What does! mean (and give the formula)? n object has a velocity of 5 m/s and starts 0 m away from you. ) How far does it travel each second? B) Where is it after 1 second? C) Where is it after seconds? D) Where is it after 5 seconds? E) How far does it travel between seconds 7 and 8? car travels 35 m in 5 secs. Calculate its speed. Which has the faster speed? Car or Car B? Both go the same distance, but Car B gets there sooner. In the same amount of time, Car goes farther. T = T B, but D < D B. Car 1 is going 0 m/s. Car is going 30 m/s. Which one travels 0 m first? Which one can travel a greater distance? Which one travels farther in more time? B Variables: Formula: Variables: Solution: bike goes 1 m/s for 6 seconds. Calculate how far the bike traveled. Solution: C Formula: D E Choose which of the above object s motion applies to the following (can be more than one): For the following problems, show all work and steps. plane stops from 300 mph in 15 seconds. Calculate the planes acceleration. V i = 0 Decelerating Constant speed Is stopping Positive acceleration t constant velocity V f = 0 ccelerating cceleration = 0 Distance is increasing Starts at rest Constant direction Negative acceleration V i = V f bike going 3 m/s ends up going 9 m/s after seconds. Calculate the bike s acceleration. For object B above: ) If there is 1 second between each dot, when did the object reach 1 m? B) Find the speed of object B. Speed (S) or Velocity (V) car travels m/s left. bird flies 0 m/s. bike goes m/s toward town. Scalar (S) or Vector (V) m/s. 60 mph toward ustin. Direction matters.
Unit 6:5 Speed Speed vs. Time B C D Which graph segments fit the following: Constant speed: Negative acceleration: Positive cceleration: Slowing down: Position Position vs. Time C B D Which graph segments fit the following: t rest: Fast speed: Slow speed: Going backwards: Time cceleration = 0: Time Going forward: Position (m) Position vs. Time 0 Line 18 16 14 1 8 6 4 0 0 1 3 4 5 6 Time (sec) n object accelerates at m/s. nswer the following: ) If it starts at rest, how fast is it going after 1 second? Which is the independent variable? Which is the dependent variable? Where was the object at 4 seconds? Where did the object start? When did the object reach 8 meters? Find the slope of the graph (show work) What does the slope you just found stand for? If two objects have a net momentum of 45 kgm/s before they collide, how much momentum will they have after they collide? B) fter seconds, how fast is it going? C) If it starts at 5 m/s, how fast would it be going after 1 second? n astronaut is by herself in space. ll she has is a box of tools. How can she get to her ship that is to her left? For the following problems, show all work and steps. 4 kg object is moving 6 m/s to the left. Calculate momentum. How is it possible that two moving objects can collide and stop moving? kg object has 58 kgm/s of momentum. Find its velocity. Find the net momentum of the two objects at the right. 4 m/s kg 3 m/s 8 kg 00 kg cannon shoots a kg cannonball. If the ball ends up going 300 m/s to the right: ) If they are both at rest beforehand, what is "p before? B) What is "p after? C) Is the ball s final p positive or negative (p ball )? D) Is the cannon s final p positive or negative (p cannon )? E) Find the velocity of the cannon afterwards v cannon )? Number these from most (1) to least (5) momentum. Fast car Fast truck Fast plane Fast hammer mountain