10. Physics Applications of Integrals

Acceleration A drag racer accelerates at a(t)=88 ft/s². Assume v(0)=0, s(0)=0, and t is measured in seconds.

c. At this rate, how long will it take the racer to travel 1/4 mi?

29–36. Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. Use the Fundamental Theorem of Calculus (Theorems 6.1 and 6.2).

a(t) = cos2t; v(0) = 5; s(0) = 7

29–36. Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. Use the Fundamental Theorem of Calculus (Theorems 6.1 and 6.2).

a(t) = e^−t; v(0) = 60; s(0) = 40

29–36. Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. Use the Fundamental Theorem of Calculus (Theorems 6.1 and 6.2).

a(t) = −32; v(0)=50; s(0)=0

Probe speed A data collection probe is dropped from a stationary balloon, and it falls with a velocity (in m/s) given by v(t) = 9.8t, neglecting air resistance. After 10 s, a chute deploys and the probe immediately slows to a constant speed of 10 m/s, which it maintains until it enters the ocean.

c. If the probe was released from an altitude of 3 km, when does it enter the ocean?

Probe speed A data collection probe is dropped from a stationary balloon, and it falls with a velocity (in m/s) given by v(t) = 9.8t, neglecting air resistance. After 10 s, a chute deploys and the probe immediately slows to a constant speed of 10 m/s, which it maintains until it enters the ocean.

b. How far does the probe fall in the first 30 s after it is released?

Day hike The velocity (in mi/hr) of a hiker walking along a straight trail is given by v(t) = 3 sin² πt/2, for 0≤t≤4. Assume s(0)=0 and t is measured in hours. 

c. What is the hiker’s position at t=3?

Flying into a headwind The velocity (in mi/hr) of an airplane flying into a headwind is given by v(t) = 30(16−t²), for 0≤t≤3. Assume s(0)=0 and t is measured in hours.

c. How far has the airplane traveled at the instant its velocity reaches 400 mi/hr?

Cycling distance A cyclist rides down a long straight road with a velocity (in m/min) given by v(t) = 400−20t, for 0≤t≤10, where t is measured in minutes.

c. How far has the cyclist traveled when her velocity is 250 m/min?

Cycling distance A cyclist rides down a long straight road with a velocity (in m/min) given by v(t) = 400−20t, for 0≤t≤10, where t is measured in minutes.

b. How far does the cyclist travel in the first 10 min?

17–22. Position from velocity Consider an object moving along a line with the given velocity v and initial position.

a. Determine the position function, for t≥0, using the antiderivative method

v(t) = 9−t² on [0, 4]; s(0)=−2

17–22. Position from velocity Consider an object moving along a line with the given velocity v and initial position.

a. Determine the position function, for t≥0, using the antiderivative method

v(t) = 6−2t on [0, 5]; s(0)=0

17–22. Position from velocity Consider an object moving along a line with the given velocity v and initial position.

a. Determine the position function, for t≥0, using the antiderivative method

v(t) = −t³+3t²−2t on [0, 3]; s(0)=4

13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.

b. Find the displacement over the given interval. 

v(t) = 50e^−2t on [0, 4]

13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.

a. Determine when the motion is in the positive direction and when it is in the negative direction. 

v(t) = 50e^−2t on [0, 4]