Kinematics and Calculus

Problems

practice

Determine the equations of motion for constant jerk.
An object's position is described by the following polynomial for 0 to 10 s.
x = t³ − 15t² + 54t

Where x is in meters, t is in seconds, and positive is forward. Determine …
1. the object's velocity as a function of time
2. the object's acceleration as a function of time
3. the object's maximum velocity
4. the object's minimum velocity
5. the time when the object was moving backward
6. the times when the object returned to its starting position
7. the object's average velocity
8. 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.

[magnify]
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 …
1. the maximum speed of the elevator
2. the duration of the brief jerk experienced by the elevator centered on 17.5 s
Sketch the corresponding graphs of …
1. velocity-time
2. displacement-time
Determine …
1. the most likely floor on which the elevator stopped

algebraic

Determine the acceleration–velocity 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 …
1. the object's position, s, as a function of time
2. the object's acceleration, a, as a function of time
3. the object's maximum velocity
4. if and when when the object stops
5. 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, x₀, π and τ are all constants.)
1. x = ⅙jt³
2. x = A sin(2πft)
3. x = x₀e^−t/τ
A crude mathematical model of tunneling is represented by the equation …

v = k

x

where v is the tunneling speed; x is the length of the tunnel; and k is a constant.
1. 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?
2. Determine the following quantities as a function of time …
  1. tunnel length
  2. tunneling speed
  3. tunneling acceleration
  (This is a somewhat difficult problem for students who have just started learning calculus.)
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.
1. speed
  1. What are the units in the coefficients A and b?
  2. What is the physical meaning of the coefficent A?
  3. What is the speed of the car at t = 0 s?
  4. What is the asymptote of this function as t → ∞?
  5. Sketch a graph of speed vs. time. Include the value of v(0) and the asymptote of v as t → ∞.
2. position
  1. Derive an equation x(t) for the instantaneous position of the car as a function of time. (Be sure that your function has the value x = 0 when t = 0.)
  2. What is the asymptote of this function as t → ∞?
  3. What is the physical meaning of the slope of this asymptote?
  4. Sketch a graph of position vs. time. Include the value of x(0) and the asymptote of x as t → ∞.
3. acceleration
  1. Derive an equation a(t) for the instantaneous acceleration of the car as a function of time.
  2. What is the acceleration of the car at t = 0 s?
  3. What is the asymptote of this function as t → ∞?
  4. Sketch a graph of acceleration vs. time. Include the value of a(0) and the asymptote of a as t → ∞.
4. numbers
  
  source: Fifth Gear
  Apply this model to a real but exceptional car — the Red Victor 1. This car has a zero-to-sixty time of about one second and a quarter mile time of about eight seconds. In other words, let …
  
  v(1 sec) = 88 ft/sec and x(8 sec) = 1320 ft
  
  then determine …
  1. the values of the coefficients A and b [I think this can only be done using a fancy calculator.]
  2. the maximum speed, and
  3. the maximum acceleration.

statistical

elevator.txt
The acceleration-time 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.
1. Velocity
  1. Construct a velocity-time graph.
  2. Detemine the cruising speed of the elevator.
2. Displacement
  1. Construct a displacement-time graph.
  2. Determine the height of the building.
  3. Estimate the number of floors in the building.
splits.html
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.
1. What were the runner's initial and final speeds?
2. What was the runner's maximum speed and when did it occur?
3. What was the runner's average speed?
4. Did the runner's speed increase, decrease, or remain roughly the same near the end of the race?
5. 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?

timeslips.html
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.

The webpage that accompanies this problem has links to images of 50 different time slips — a sampling of the thousands compiled by enthusiasts at dragtimes.com.

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.)

distance			left car time (s)		right car time (s)

Drag Racing Time Slip — Raw Data
(feet)	(miles)	(m)	elapsed	race	elapsed	race
0	0	0	0		0
60	–	18
330	–	101
660	⅛	201
1000	–	305
1320	¼	402

Plot the distance–time data for each car. (Make two graphs.)
Perform the following curve fits on each graph …
1. linear
2. quadratic
3. 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 (t⁰, t¹, t², t³), how much scaling (× 2, × 3, × 4, … , ÷ 2, ÷ 3, ÷ 4, … ) it took to get your answer.

quantity	methodology	left car	right car

Drag Racing Time Slip — Analysis
average speed
initial speed
average acceleration
initial acceleration
average jerk

The Physics Hypertextbook

No condition is permanent.


v(1 sec) = 88 ft/sec	and	x(8 sec) = 1320 ft