Graphs of Motion

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 1,000 s to 2,000 s?
6. What was the vertical acceleration of the balloons from 0 s to 1,000 s? (Assume the acceleration was constant)
7. What was the vertical velocity of the balloons from 3,300 s to 4,300 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 timeelapsed in hours.
      2. Use the fact that the numbered streets in Manhattan are spaced 20 per mile and determine the distancetraveled 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.txt
   The Indian Space Research Organisation (ISRO) launched the Mars Orbiter Mission from the Satish Dhawan Space Centre in Andhra Pradesh on 5 November 2013. The orbiter has been given the nickname मंगलयान (Mangalyaan ), which is Sanskrit for "Mars craft". It was the 25th ISRO flight to use their four-stage Polar Satellite Launch Vehicle, thus the mission number PSLV‑C25. The table below gives velocity-time data at significant moments of the launch adapted from an ISRO document. Use this data to solve the following problems.
   1. Construct 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.
   3. Using numerical integration, create a distance-time graph from first stage ignition to spacecraft separation.

   For the more advanced student.

   1. Fit an exponential approach curve to the velocity-time 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.
   PSLV‑C25/Mars Orbiter launch events ☞ This table is also available as a tab-delimited text file.

   event	time (s)	velocity (m/s)
   first stage & ground lit strap‑on ignition	000 0.00	000 0.00
   air lit strap‑on ignition	00 25.04	0 611.52
   ground lit strap‑on separation	00 70.04	1434.17
   air lit strap‑on separation	00 92.04	2024.36
   first stage separation	0 112.75	2387.67
   second stage ignition	0 112.95	2387.16
   closed loop guidance initiation	0 117.95	2415.46
   heat shield separation	0 201.75	3624.69
   second stage separation	0 264.74	5379.33
   third stage ignition	0 265.94	5378.94
   third stage separation	0 583.60	7730.88
   fourth stage ignition	2100.50	7642.04
   fourth stage burn out	2619.72	9833.49
   spacecraft separation	2656.72	9804.01

   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.

6. mustang-velocity.pdf
   In 2016 Road & Track 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 scatter plot of speed (in meter per second) vs. time (in seconds), then add a best fit curve.
7. 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 scatter plot 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?
8. 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.
9. David 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 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 1,000 s to 2,000 s?
   6. What was the vertical acceleration of the balloons from 0 s to 1,000 s? (Assume the acceleration was constant)
   7. What was the vertical velocity of the balloons from 3,300 s to 4,300 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?
   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?

   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.)