AP Calculus AB: 10.1.1 Antiderivatives and Motion
This section explains how calculus is used to model motion, showing how antiderivatives of acceleration and velocity functions help determine velocity and position over time. It highlights solving initial value problems, interpreting when an object stops, and applying these principles in real-life motion scenarios.
Antiderivatives and Motion
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
Key Terms
Antiderivatives and Motion
Position and motion can be analyzed using calculus.
• Velocity is the rate of change of position with respect to time.
Acceleration is the ra...
note
Integral calculus empowers you to take an acceleration function and deduce the velocity and position functions.
In this exam...
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.
v (t) = 8t ^2 − 2t^ 3 − 3
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?
19,200 feet
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.
p (t) = −300 sin t − 140 cos t + 350t + 140
Find the antiderivative, F (x), of f (x) = 8x ^3 + 6x that satisfies the condition F (1) = 6.
F (x) = 2x^4 + 3x ^2 + 1
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| Term | Definition |
|---|---|
Antiderivatives and Motion | Position and motion can be analyzed using calculus. |
note |
|
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. | v (t) = 8t ^2 − 2t^ 3 − 3 |
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? | 19,200 feet |
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. | p (t) = −300 sin t − 140 cos t + 350t + 140 |
Find the antiderivative, F (x), of f (x) = 8x ^3 + 6x that satisfies the condition F (1) = 6. | F (x) = 2x^4 + 3x ^2 + 1 |
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? | 49 meters / sec |
What constant acceleration is needed to accelerate a baseball from 6 feet / sec to 116 feet / sec in 2 seconds? | 55 feet / sec^2 |
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 ? | p(t)=A/2t^2+v0t+p0 |
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? | 90 miles |
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? | 426.4 feet / sec |
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? | 8.4 seconds |