Motion and Antiderivatives None of what you’ll find in these notes is new, but it’s important to pull it all together to make sure we can think about motion not just in terms of derivatives, but also in terms of antiderivatives. Problem: A particle moves along the y-axis and has position y = 3 when t = 0 . A graph of the particle’s velocity as a function of t is shown below.

a. Determine the particle’s speed at t = 6 . Is the speed increasing, decreasing or constant at that time?

b. At what time(s) does the particle reach the origin?

c. Find the absolute lowest and absolute highest points reached by the particle. (Absolute max/min, eh? What test is this going to involve?)

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Problem: (Calculator) A particle moves along a straight line. For 0 ≤ t ≤ 3 , the velocity of the particle is 5/3 1 given by v ( t ) = 2 + ( t 3 − 4t ) − t 2 , and the position of the particle is given by s ( t ) . It is given that 5 s (0) = 3 . a. Find all values of t in the interval 0 ≤ t ≤ 3 for which the speed of the particle is 1.

b. Write an expression involving an integral that gives the position s ( t ) . Use this expression to find the position of the particle at t = 2 .

c. Find all times t at which the particle changes direction. Justify your answer.

d. Is the speed of the particle increasing or decreasing at time t = 1 ? Give a reason for your answer.

Problem: The velocity, in miles per minute, of an object moving along a horizontal axis is modeled by the piecewise-linear function whose graph is shown below.

a. Find the acceleration at t = 1.5 minutes. Indicate units of measure.

b. Using correct units, find the value and explain the meaning of

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∫ v (t ) dt . 10

0

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t 0 6 18 24 30 v (t ) 5 -4 -8 -1 6 Problem: The velocity of a particle moving along the x-axis is modeled by a differentiable function v, where x is measured in feet, and t is measured in seconds. Selected values of v ( t ) are given in the table above. The particle’s position is x = 4 feet when t = 0 . a. Estimate the acceleration of the particle at t = 12 seconds. Show the computations that lead to your answer. Indicate units of measure.

b. Using correct units, explain the meaning of

∫ v (t ) dt 24

6

in the context of the problem. Use a

trapezoidal sum with two subintervals indicated by the data in the table to approximate

∫ v (t ) dt . 24

6

c. For 0 ≤ t ≤ 30 , must the particle change direction in any of the subintervals indicated by the data in the table? If so, identify the subintervals and explain your reasoning. If not, explain why not.

d. Suppose that acceleration of the particle is negative for 0 < t < 6 seconds. Explain why the particle’s position at t = 6 must be less than x = 35 feet.

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Problem: For 0 ≤ t ≤ 6 , a particle is moving along the y-axis. The particle’s position, y ( t ) , is not

( )

explicitly given. The velocity of the particle is given by v ( t ) = −3cos et 3 − 1 . The acceleration of the

( )

particle is given by a ( t ) = et 3 sin et 3 and y ( 0 ) = 3 .

a. Is the speed of the particle increasing or decreasing at t = 4 ? Give a reason for your answer.

b. Find the average velocity of the particle on the time interval 0 ≤ t ≤ 6 .

c. Find the total distance traveled by the particle on the time interval 0 ≤ t ≤ 6 .

d. Does the particle ever reach the origin on the time interval 0 ≤ t ≤ 6 ?

e. Find the particle’s maximum acceleration on the time interval 0 ≤ t ≤ 6 .

f. Use the line tangent to y ( t ) at t = 0 to estimate y (1) . Is this an over or underestimate?

g. Write an expression involving an integral that gives the position y ( t ) . Use this expression to find y (1) .

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