10.1.1 Antiderivatives and Motion

Question 1

Antiderivatives and Motion

Answer 1

• Position and motion can be analyzed using calculus.
• Velocity is the rate of change of position with respect to time.
Acceleration is the rate of change of velocity with respect to time.
• Given the velocity function of an object and its position at a specific time, find its position function by taking the antiderivative
of velocity and solving for the specific constant of integration.
• Given the acceleration function of an object and its velocity at a specific time, find its velocity function by taking the antiderivative of acceleration and solving for the specific constant of integration.
• An object stops moving when its velocity becomes zero

Question 2

note

Answer 2

• Integral calculus empowers you to take an acceleration function and deduce the velocity and position functions.
• In this example, Professor Burger states that he was riding his bicycle at 30 ft/sec when he put on the brakes for a constant deceleration of 20 ft/sec 2 . Express deceleration as negative acceleration.
• Prof. Burger’s initial velocity v(0) was 30 ft/sec and his initial position p(0) was 0 ft.
• Since acceleration is the derivative of velocity, you can integrate the acceleration function to find the velocity function. You can determine the value of C by using the initial velocity value of 30 ft/sec.
• To determine the position function, integrate the velocity function. Once again, use the initial condition you know for position.
• Before you can determine the distance Prof. Burger traveled before stopping, you need to know how much time elapsed. The bicycle has come to a stop when the velocity equals 0, so set the velocity function equal to 0 and solve for time t.
• Evaluate the position function for the time of 3/2 sec to arrive at the final position. Since the initial position was defined to be zero, the final position is also the distance traveled, 22.5 ft.
• The velocity function is a function expressing a rate of change. The position function evaluated at 3/2 sec, obtained by integrating the velocity function, gives the accumulated change in Professor Burger’s position from time 0 to time 3/2 sec.

Question 3

A model rocket blasts off and experiences an acceleration described by the function a (t) = 16t − 6t ^2, where a (t) is in meters / sec2. Find the function which describes the velocity of the rocket if it is moving upwards at 3 meters / sec at t = 1.

Answer 3

v (t) = 8t ^2 − 2t^ 3 − 3

Question 4

A supersonic jet accelerates at a constant rate of 150 feet / sec2 until it reaches its maximum velocity of 1200 feet / sec. What is the maximum distance the jet can travel in 20 seconds if it starts from rest?

Answer 4

19,200 feet

Question 5

The “Back and Forth” thrill ride moves with an acceleration function of a (t) = 300 sin t + 140 cos t. Find the position of the ride after t seconds if its initial velocity is 50 and its initial position is 0.

Answer 5

p (t) = −300 sin t − 140 cos t + 350t + 140

Question 6

Find the antiderivative, F (x), of f (x) = 8x ^3 + 6x that satisfies the condition F (1) = 6.

Answer 6

F (x) = 2x^4 + 3x ^2 + 1

Question 7

Ken is driving down the road when a car suddenly pulls out in front of him. He applies the brake sharply and his car goes into a skid. While the car skids, it decelerates at a constant rate of 15 meters / sec2. If the car skids for 80 meters before stopping, how fast was Ken driving before he hit the brakes?

Answer 7

49 meters / sec

Question 8

What constant acceleration is needed to accelerate a baseball from 6 feet / sec to 116 feet / sec in 2 seconds?

Answer 8

55 feet / sec^2

Question 9

Suppose that an object is moving in a straight line with a constant acceleration a (t) = A. If the initial velocity of the object is v0, and the initial position of the object is p0, which of the following functions gives the position of the object at time t ?

Answer 9

p(t)=A/2t^2+v0t+p0

Question 10

Shana rides along a straight path with a velocity given by the equation v (t) = 5t ½, where t is given in hours and the velocity is in miles per hour. If Shana starts her journey at t = 0, how far has Shana traveled after 9 hours?

Answer 10

90 miles

Question 11

A dragster on its way down a 1200 ft course has an acceleration function a (t) = 12t, where t is the time in seconds and a (t) is measured in ft / sec2. If the dragster starts the race from a standstill at the beginning of the course at t = 0, how fast is the dragster going when it crosses the finish line?

Answer 11

426.4 feet / sec

Question 12

A dragster on its way down a 1200 foot course has an acceleration function of a (t) = 12t, where t is the time in seconds and a (t) is in feet / sec2. If it starts the race from a standstill at t = 0, how long does it take the dragster to finish the race?

Answer 12

8.4 seconds