The Physics
Hypertextbook
Opus in profectus

# Graphs of Motion

## Problems

### practice

1. Complete the table on the first page of worksheet-compare.pdf. Fill each grid space with an appropriately concise answer.
2. worksheet-transform.pdf
The graph below shows velocity as a function of time for some unknown object. 1. What can we say about the motion of this object?
2. Plot the corresponding graph of acceleration as a function of time.
3. Plot the corresponding graph of displacement as a function of time.
3. Sketch the displacement-time, velocity-time, and acceleration-time graphs for…
1. an object moving with constant velocity. (Let the initial displacement be zero.)
2. an object moving with constant acceleration. (Let the initial displacement and velocity be zero.)
4. The graph below shows the altitude of a skydiver initially at rest as a function of time. After 7 s of free fall the skydiver's chute deployed completely, which changed the motion abruptly.
1. Determine the velocity at the instant…
1. just before the parachute opened
2. just after the parachute opened
2. What was the skydiver's acceleration…
1. from the beginning of the jump to the time just before the parachute opened?
2. from the time just after the parachute opened to the time when the skydiver landed?
3. Sketch the corresponding graphs of…
1. velocity-time
2. acceleration-time ### conceptual

1. worksheet-choose-displacement.pdf
The graphs on the accompanying pdf show the displacement of a hypothetical object moving along a straight line. Choose the lettered graph that best represents each of the numbered descriptions. A graph may be used for more than one description or it may not be used at all. Some descriptions may correspond to more than one graph and some may not correspond to any graph at all.
2. worksheet-choose-velocity.pdf
The graphs on the accompanying pdf show the velocity of a hypothetical object moving along a straight line. Choose the lettered graph that best represents each of the numbered descriptions. A graph may be used for more than one description or it may not be used at all. Some descriptions may correspond to more than one graph and some may not correspond to any graph at all.
3. Sketch the displacement-time, velocity-time, and acceleration-time graphs for each of the following scenarios. (Be prepared to explain your sketches.)
1. An elevator that ascends from the lobby to the 36th floor, stops, descends to the 27th floor, stops, and returns to the lobby.
2. A basketball is dropped on the court and allowed to bounce up and down several times undisturbed.
3. A car on a test track performing a zero-to-sixty acceleration test. (This acceleration will not be uniform.)
4. A race between a tortoise and a hare that unfolds just like the fable of the same name. (An acceleration-time graph is not necessary for this particular problem.)
5. Two cars are adjacent to each other on a four-lane highway. The first car accelerates uniformly from rest the moment the light changes to green. The second car approaches the intersection already moving and is beside the first car at the instant the light changes. It then continues driving with a constant velocity.
6. Traffic lights on some streets are timed to facilitate traffic flow at a certain speed. Goofus and Gallant are stopped at a red light on this kind of street. When the light changes Goofus hammers the accelerator until he exceeds the speed limit. He arrives at the next light which is still red and stops. Gallant accelerates at a reasonable rate and never exceeds the speed limit. The light turns green at just the right instant so that Gallant does not need to brake. Goofus accelerates and brakes again. Gallant continues driving at a constant speed.

### numerical

1. worksheet-make-displacement.pdf
The worksheet for this exercise consists of three small and one large displacement-time graph.
1. Complete the three small displacement-time graphs from the information provided below each graph.
2. The larger displacement-time graph shows the motion of some hypothetical object over time. Break the graph up into segments and describe qualitatively the motion of the object in each segment. Whenever possible, calculate the velocity of the object as well.
2. worksheet-make-velocity.pdf
The worksheet for this exercise consists of three small and one large velocity-time graph.
1. Complete the three small velocity-time graphs from the information provided below each graph.
2. The larger velocity-time graph shows the motion of some hypothetical object over time. Break the graph up into segments and describe qualitatively the motion of the object in each segment. Whenever possible, calculate the acceleration of the object as well.
3. The graph below shows the vertical velocity of a skydiver as a function of time. At time t = 0 s the skydiver is located at position y = 0 m at the door of the plane, at t = 8 s the parachute opened, and at t = 12 s the skydiver touched down. Assume that the positive directions for displacement, velocity, and acceleration are downward. Using this information sketch the corresponding graphs of…
1. displacement-time
2. acceleration-time 4. The graph below shows the speed of the Shanghai Maglev train as it traveled from Pudong International Airport to Longyang Road Station. At 38 s the train began accelerating from rest. At 172 s the train reached its maximum speed of 301 km/h. At 390 s the train began decelerating. At 515 s the train stopped completely.
1. Determine the acceleration of the Shanghai Maglev…
1. as it left the first station
2. as it approached the last station
2. Determine the distance traveled by the Shanghai Maglev…
1. while it was accelerating
2. while it was cruising
3. while it was decelerating
3. Determine the distance from Pudong to Longyang.
4. Determine the average speed of the Shanghai Maglev while it was in motion.
5. Sketch the corresponding graphs of…
1. distance vs. time
2. acceleration vs. time Adapted from Tung Hsu, 2014.
5. The graph below shows the position of a car pulling into a parking spot as a function of time. Use this graph to answer the following questions. The positive direction is forward. (Signs matter.) 1. Determine the velocity of the car from 0 to 1 seconds.
2. Determine the velocity of the car from 3 to 4 seconds.
3. Determine the velocity of the car from 8 to 9 seconds.
4. Determine the acceleration of the car from 1 to 3 seconds.
5. Determine the acceleration of the car from 4 to 8 seconds.
6. Determine the displacement of the car from 1 to 3 seconds.
7. Determine the displacement of the car from 3 to 4 seconds.
8. Determine the displacement of the car from 4 to 8 seconds.
9. Sketch the corresponding velocity-time graph
10. Sketch the corresponding acceleration-time graph
6. The graph below shows the velocity of a car pulling into a parking spot as a function of time. Use this graph to answer the following questions. The positive direction is forward. (Signs matter.) 1. Determine the velocity of the car from 0 to 1 seconds.
2. Determine the velocity of the car from 3 to 4 seconds.
3. Determine the velocity of the car from 8 to 9 seconds.
4. Determine the acceleration of the car from 1 to 3 seconds.
5. Determine the acceleration of the car from 4 to 8 seconds.
6. Determine the displacement of the car from 1 to 3 seconds.
7. Determine the displacement of the car from 3 to 4 seconds.
8. Determine the displacement of the car from 4 to 8 seconds.
9. Sketch the corresponding position-time graph
10. Sketch the corresponding acceleration-time graph
7. The velocity-time graph below was derived from a video of a student jumping rope (a single jump). Use this graph and your knowledge of physics to answer the the following questions.
1. At what times was the student momentarily, and only momentarily, at rest?
2. When was the student's speed the greatest?
3. What was the speed of the student at the time you identified in part b?
4. Over what time interval was the student's acceleration the greatest?
5. Calculate the acceleration of the student during the interval you identified in part d.
6. Calculate the displacement of the student from 0.367 s to 0.745 s. 8. youtu.be/QwzvNAAqH3gDavid Blaine is an American performer famous for stunts involving extreme endurance. In 2020, he strapped himself to a "bunch of helium balloons" and floated up to an altitude where he needed an external oxygen tank to breathe. He then detached himself from the balloons and and parachuted back to the Arizona desert. Although the appearance of the balloon bundle made it look like Mr. Blaine was heading off to a party, he was actually flying a civil aircraft. This meant obtaining a balloon pilot license, an aircraft registration code (N947DB), and an air traffic control transponder. The last of these was the source of the data for this problem. Using the altitude-time graph on the accompanying PDF file (ascension.pdf) and your knowledge of physics, answer the following questions.
1. What was the altitude of the balloons when they were launched?
2. What was the altitude of the balloons when they landed?
3. Where and when were the balloons at their greatest altitude?
1. At what altitude in meters?
2. At what time in seconds?
4. Where and when did David Blaine release the balloons and begin his dive?
1. At what altitude in meters?
2. At what time in seconds?
5. What was the vertical velocity of the balloons from 1000 s to 2000 s?
6. What was the vertical acceleration of the balloons from 0 s to 1000 s? (Assume the acceleration was constant)
7. What was the vertical velocity of the balloons from 3300 s to 4300 s?

Answer the final two questions using words, not numbers. Do not do any calculations.

1. When did the balloons have their greatest upward speed?
2. When did the balloons have their greatest downward speed?

### statistical

1. take-the-a-train.txt
The A Train makes the longest run of any subway in the New York City Transit system. The stretch from 207 Street to Broadway-Nassau is just about as long as the entire island of Manhattan. The data in the accompanying text file were taken from the 2008 weekday schedule for the A Express Train.
1. Add two new columns to the data table.
1. Use the time of day given in the timetable to determine the time elapsed in hours.
2. Use the fact that the numbered streets in Manhattan are spaced 20 per mile and determine the distance traveled in miles.
2. Construct a distance-time graph with a line of best fit and use it to determine the following quantities in Anglo-American units…
1. the average speed of the A Train.
2. the length of Manhattan.
3. the length of the A line.
2. jet-takeoff.txt, jet-landing.txt
One fine day, a Boeing 717 departed from Mitchell International Airport (MKE) in Milwaukee. Approximately two hours later, it arrived at LaGuardia Airport (LGA) in New York. During takeoff and landing, runway positions (in meters) were recorded as a function of time (in seconds) and the data were saved as tab-delimited text files. Using the data in these files and your favorite graphing software…
1. construct a graph of distance vs. time for…
1. takeoff and
2. landing
2. then fit a quadratic curve to the data so that you can determine…
1. the acceleration at takeoff and
2. the deceleration on landing
3. and also determine…
1. the final speed when the airplane left the runway in Milwaukee and
2. the initial speed when the airplane hit the runway in New York
3. A picket fence is a type of fence (obviously). This kind of fence is made out of evenly spaced, vertically aligned, pointed slabs of wood tied together near the top and bottom by cross members. A picket fence is also the name of a piece of laboratory equipment used by introductory physics students. This kind of "fence" is a transparent piece of plastic with opaque bands spaced evenly across it. When this kind of picket fence passes through a photogate, the opaque and transparent bands can be used to determine position as a function of time. The second kind of picket fence was used for two experiments. Use the position-time data from each experiment to determine the acceleration due to gravity on the surface of the Earth.
1. picket-fence-falling.txt
In the first experiment, the picket fence was allowed to fall freely downward through the photogate.
2. picket-fence-rising.txt
In the second experiment, the picket fence was given a quick tap upward and then released to travel freely upward through the photogate.
4. hawaiian-chain.txt
The Hawaiian Island chain is more than just the visible islands. It also includes the Emperor Seamounts. (Seamounts are islands that have eroded down below sea level.) The combined Hawaii-Emperor chain is a series of volcanic structures formed by a single, long-lived plume of magma referred to as a "hotspot". The hotspot stayed fixed as the pacific plate slowly moved over it, resulting in a chain of volcanoes stretching from the Aleutian Islands off the coast of Alaska to Mount Kilauea on the Big Island of Hawaii. Use this data to determine the speed of the Pacific plate. The columns in this data set are as follows:
1. Volcano number
2. Volcano name
3. Volcano age (millions of years)
4. Distance from Kilauea (km)
5. Uncertainty in age (millions of years)
6. Uncertainty in distance (km)
Data source: Clague, Dalrymple, et al. 1989
5. pslv-c25.txtThe Indian Space Research Organisation (ISRO) launched the Mars Orbiter Mission from the Satish Dhawan Space Centre in Andhra Pradesh on 5 November 2013. The Mars Orbiter Spacecraft has been given the nickname मंगलयान (transliterated to Maṅgalayāna or Mnglyan), which is Sanskrit for "Mars craft". English speaking news agencies have been writing this as "Mangalyaan".

The launch system used was an "extended" version of the four-stage Polar Satellite Launch Vehicle (PSLV-XL) built by ISRO. The launch number for the Mars Orbiter Mission was C25. The accompanying text file gives velocity-time data at significant moments of the PSLV–C25 launch taken from a page on the ISRO website. Use this data to solve the following problems.

1. Contruct a velocity-time graph of the launch.
2. Using a line of best fit, determine the average acceleration from first stage ignition to third stage separation (i.e., the portion of the data set where the acceleration is most nearly uniform).
3. Using numerical integration, create a distance-time graph from first stage ignition to spacecraft separation (i.e., the whole data set).

For the more advanced student.

1. Fit an exponential approach curve to the data from first stage ignition to third stage separation (the same range of values used for part b of this problem).
2. Using the results of your curve fit, derive an expression for acceleration as a function of time.
6. nyan.txt
Nyan Cat: Lost In Space is a game based on the 2011 internet meme of a Pop-Tart-cat hybrid that leaves a flowing rainbow trail behind him. In the game, he runs across the screen, jumping between horizontal sausages floating in space. A players gains points every time Nyan Cat manages to catch a piece of food (candy, cake, donuts, ice cream, or milk) or something else valuable (coins or jewels). If Nyan Cat doesn't land on a sausage he falls to his death and the game is over. At the end of a run, the player's score is displayed along with the distance and duration of the run. The game was played nine times and the results were recorded in nyan.txt. Determine the speed of Nyan Cat in this game using this data and graphical methods.
7. mustang-velocity.pdf
In 2016 Road & Track (paid link) magazine tested eight very expensive and very fast cars to determine the Performance Car of the Year. Data from an acceleration test for a 2016 Ford Mustang Shelby GT350R are given in the table below. (The Mustang did not win the award that year.) Since the data were collected in the United States, the milestone speeds were chosen as multiples of 10 mph. For your convenience, these speeds were converted to SI units. In the space below, make a scatterplot of speed (in meter per second) vs. time (in seconds), then add a best fit curve.
8. mustang-acceleration.pdf
Using the data from the previous worksheet, complete the table below. For every interval on the previous table compute the speed change (in meter per second), the time change (in seconds), the acceleration (in meter per second squared), and the average time (in seconds). In the space below, make a scatterplot of average acceleration vs. average time, then add a best fit curve. Finally, answer the following three qualitative questions.
1. During this test, did the distance traveled by the car increase, decrease, or remain the same?
2. During this test, did the speed of the car increase, decrease, or remain the same?
3. During this test, did the acceleration of the car increase, decrease, or remain the same?
9. jump-rope.pdf
The velocity-time graph in the middle of this worksheet was derived from a video of a student jumping rope (a single jump). Construct the corresponding position-time and acceleration-time graphs.
10. youtu.be/QwzvNAAqH3gDavid Blaine is an American performer famous for stunts involving extreme endurance. In 2020, he strapped himself to a "bunch of helium balloons" and floated up to an altitude where he needed an external oxygen tank to breathe. He then detached himself from the balloons and and parachuted back to the Arizona desert. Although the appearance of the balloon bundle made it look like Mr. Blaine was heading off to a party, he was actually flying a civil aircraft. This meant obtaining a balloon pilot license, an aircraft registration code (N947DB), and an air traffic control transponder. The last of these was the source of the data for this problem. Using the altitude-time data in the accompanying tab delimitted text file (ascension.txt) and your favorite application for analyzing data, answer the following questions.
1. What was the altitude of the balloons when they were launched?
2. What was the altitude of the balloons when they landed?
3. Where and when were the balloons at their greatest altitude?
1. At what altitude in meters?
2. At what time in seconds?
4. Where and when did David Blaine release the balloons and begin his dive?
1. At what altitude in meters?
2. At what time in seconds?
5. What was the vertical velocity of the balloons from 1000 s to 2000 s?
6. What was the vertical acceleration of the balloons from 0 s to 1000 s? (Assume the acceleration was constant)
7. What was the vertical velocity of the balloons from 3300 s to 4300 s?

Answer the final two questions using words, not numbers.

1. When did the balloons probably have their greatest upward speed?
2. When did the balloons probably have their greatest downward speed?
11. bolt.txt
A split is a time at which the runner reaches a milestone distance in a race. In the 100 m dash, for example, split times are taken every 10 m. Splits for three of Jamaican sprinter Usain Bolt's best races are given on the accompanying text file. Pick any one of these races and complete the following tasks.
1. Compute the average speed during each of the ten 10 m intervals.
2. Construct a speed-time graph.
3. Identify the 10 m interval with the greatest average speed.
4. Did Mt. Bolt's speed decrease, increase, or remain the same at the end of the race?
Data adapted from Maćkała and Mero, 2013.

### investigative

1. The numbered streets in Manhattan above 14th Street are spaced apart such that twenty blocks equal one mile. Ride one of the local trains that runs beneath an avenue for at least five consecutive stations. Using a timer or a wristwatch record the starting and stopping times of the train and the street number of the station until you have reached the fifth station. Translate your data into a displacement-time and velocity-time graph. Include the necessary data tables. Use whatever units you wish. (This investigation can also be performed in other places in a car or a bus if the streets are gridded and you know the grid interval.)