Kinematics and Calculus
Problems
practice
 Derive the equations of motion for constant jerk.
 An object's position is described by the following polynomial for 0 to 10 s…
s = t^{3} − 15t^{2} + 54t
where s is in meters, t is in seconds, and positive is forward. Determine…
 the object's velocity as a function of time
 the object's acceleration as a function of time
 the object's maximum velocity
 the object's minimum velocity
 the time when the object was moving backward
 the times when the object returned to its starting position
 the object's average velocity
 the object's average speed
 The graph below shows the acceleration of a hydraulic elevator in a four story school building as a function of time.
The graph begins at t = 0 s when the elevator door closed on the second floor and ends at t = 20 s when the door opened on a different floor. Assume that the positive directions for displacement, velocity, and acceleration are upward. Determine…
 the maximum speed of the elevator
 the duration of the brief jerk experienced by the elevator centered on 17.5 s
 velocitytime
 positiontime
 the most likely floor on which the elevator stopped
 One fine day on an unused airport runway, a highend sports car conducted a 0 to 400 km/h performance test. Its velocity changed according to the following function…
v = a(1 − e^{−t/b})
where…
a = 128.1 m/s b = 13.31 s Answer these three related questions.
 How long did it take the car to reach 400 km/h (111.111 m/s)?
 What was its average acceleration during the test?
 What is the car's theoretical top speed?
Answer these three related questions.
 Derive an expression for acceleration as a function of time.
 What was the acceleration of the car when the test started?
 What was the acceleration of the car when it hit 400 km/h?
Answer these two related questions.
 Derive an expression for displacement as a function of time.
 What distance did the car travel while accelerating?
After reaching the target speed of 400 km/h (111.111 m/s), the driver immediately disengaged the engine and applied the brakes. The car came to a complete stop after 9.451 s. Answer these three related questions.
 What was the average acceleration of the car while stopping?
 What distance did the car travel while stopping?
 What total distance did the car travel from start to finish.
algebraic
 Determine the accelerationvelocity relationship for constant jerk. (For the sake of argument, let's call this the fifth equation of motion.)
calculus
 An object's velocity, v, in meters per second is described by the following function of time, t, in seconds for a substantial length of time…
v = 4t(4 − t) + 8
Assuming the object is located at the origin (s = 0 m) when t = 0 s determine…
 the object's position, s, as a function of time
 the object's acceleration, a, as a function of time
 the object's maximum velocity
 if and when when the object stops
 if and when the object returns to the origin (s = 0 m)
 The following equations state displacement as a function of time. Derive the subsequent equations for velocity and acceleration as functions of time. (The symbols A, f, j, k, s_{0}, π and τ are all constants.)
 s = ⅙jt^{3}
 s = A sin(2πft)
 s = s_{0}e^{−t/τ}
 A crude mathematical model of tunneling is represented by the equation…
v = k s  In what way (increase or decrease) does the tunneling speed change as the tunnel gets longer? What engineering aspect of tunneling is causing this change?
 Determine the following quantities as a function of time…
 tunnel length
 tunneling speed
 tunneling acceleration
 A simplified model of a car accelerating from rest along a straight path is given by the following equation…
v(t) = a(1 − e^{−bt})
Where v(t) is the instantaneous speed of the car in feet per second, t is the time in seconds, and A and b are constants. speed
 What are the units in the coefficients a and b?
 What is the physical meaning of the coefficent a?
 What is the speed of the car at t = 0 s?
 What is the asymptote of this function as t → ∞?
 Sketch a graph of speed vs. time. Include the value of v(0 s) and the asymptote of v as t → ∞.
 position
 Derive an equation s(t) for the instantaneous position of the car as a function of time. (Be sure that your function has the value s = 0 m when t = 0 s.)
 What is the asymptote of this function as t → ∞?
 What is the physical meaning of the slope of this asymptote?
 Sketch a graph of position vs. time. Include the value of s(0 s) and the slope of the asymptote of s as t → ∞.
 acceleration
 Derive an equation a(t) for the instantaneous acceleration of the car as a function of time.
 What is the acceleration of the car at t = 0 s?
 What is the asymptote of this function as t → ∞?
 Sketch a graph of acceleration vs. time. Include the value of a(0 s) and the asymptote of a as t → ∞.

Apply this model to a real but exceptional car — the Red Victor 1. This car has a zerotosixty time of about one second and a quarter mile time of about eight seconds. In other words, let…
v(1 sec) = 88 ft/sec s(8 sec) = 1320 ft  the values of the coefficients a and b [I think this can only be done using a fancy calculator.]
 the maximum speed, and
 the maximum acceleration.
 speed
 An object's position is described by the following polynomial for 10 s…
s = t^{3} − 12t^{2} + 24t
where s is in meters and t is in seconds.
Determine…
 the object's velocity as a function of time
 the object's acceleration as a function of time
 the object's maximum velocity
 the object's minimum velocity
 the time when the object was moving backward
 the time(s) when the object was at the origin (s = 0 m)
 the time(s) when the object returned to its starting position
 the object's average velocity
 the object's average speed
statistical
 elevator.txt
The accelerationtime data in the accompanying text file were recorded by a student while riding an elevator in an office building. The student went from the lobby to the highest occupied floor. Use this data and your favorite graphing application to solve the following problems. Velocity
 Construct a velocitytime graph.
 Detemine the cruising speed of the elevator.
 Displacement
 Construct a displacementtime graph.
 Determine the height of the building.
 Estimate the number of floors in the building.
 Velocity
 tablesplits.shtml
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 some of the world's fastest sprinters are given on the accompanying webpage. Fit a high order polynomial (fourth, fifth, sixth or higher) to the data for one of these athletes using a data analysis application. Determine the speed of your sprinter as a function of time by taking the derivative of this polynomial. Graph this new function and then analyze it. What were the runner's initial and final speeds?
 What was the runner's maximum speed and when did it occur?
 What was the runner's average speed?
 Did the runner's speed increase, decrease, or remain roughly the same near the end of the race?
 How well do you think this graph describes the actual performance of the runner? Are there any problem regions on the graph? How could the function be modified to improve the fit?
 tabletimeslips.shtml
Amateur drag racing is open to anyone with a street legal vehicle (car, light truck, or motorcycle), a valid driver's license, insurance, fuel, and enough money to cover the registration fee. It is popular in the US, UK, and Australia. Races are done on a quarter mile, straight, level track. At the end of the race, each competitor is given a small paper "time slip" with data collected during the run. Data vary from venue to venue, but the following items are almost always present. Reaction time (R/T) is the time between the signal to start and when the driver actually makes the car move forward.
 Elapsed times (ET) are splits recorded at several positions. Elapsed time begins when the car crosses the starting line, not when the signal to start is given (as is done in track and field).
 Instantaneous speeds are measured at the ⅛ mile (halfway) and ¼ mile (finish). We won't use these number for this activity.
 Select one times slip and transfer the information into a table like the one below.
 Add reaction time to elapsed time to get race time. (I made up that term. I don't know what it's actually called.)
Drag racing time slip raw data distance left car time (s) right car time (s) (feet) (miles) (m) elapsed race elapsed race 0000 0 000 0 0 0060 – 018 0330 – 101 0660 ⅛ 201 1000 – 305 1320 ¼ 402  Plot the distancetime data for each car. (Make two graphs.)
 Perform the following curve fits on each graph…
 linear
 quadratic
 cubic
 Complete the following table. Be sure to include the proper units in your answers. Because you did three different curve fits, some quantities can be found by more than one method. Under "methodology" state which function (linear, quadratic, cubic), which coefficient (t^{0}, t^{1}, t^{2}, t^{3}), how much scaling (× 2, × 3, × 4, …, ÷ 2, ÷ 3, ÷ 4, … ) it took to get your answer.
quantity methodology left car right car average speed initial speed average acceleration initial acceleration average jerk Drag Racing Time Slip — Analysis
 mustang.txt
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 accompanying tabdelimited text file. (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. Using your favorite application for analyzing data, make a scatterplot of speed (in meter per second) vs. time (in seconds) and add an appropriate best fit curve.
 Use the results of your curve fit and your knowledge of calculus to create the equations for…
 acceleration as a function of time
 displacement as a function of time
 During this test…
 did the distance traveled by the car increase, decrease, or remain the same?
 did the speed of the car increase, decrease, or remain the same?
 did the acceleration of the car increase, decrease, or remain the same?
 zarmkinematics.txt
The data in the accompanying tab delimited text file give the instantaneous velocity of a vertically mounted piston used to launch projectiles as a function of time. Use this data set and your favorite application for analyzing data to solve the following problems. Using the data given, create a velocitytime graph and determine…
 the maximum velocity and
 the average velocity of the piston while it was accelerating upward
 Determine the acceleration of the piston as a function of time, create an accelerationtime graph, and determine…
 the maximum acceleration and
 the average acceleration of the piston while it was accelerating upward
 Determine the displacement of the piston as a function of time, create a displacementtime graph, and determine…
 the final displacement of the piston once it stopped moving
Data adapted from Kampen, Kaczmarczik, and Rath; 2006.
 Using the data given, create a velocitytime graph and determine…