BackProbe 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]
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.
c. Find the distance traveled over the given interval.
v(t) = 4t³ - 24t²+20t on [0, 5]
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.
c. Find the distance traveled over the given interval.
v(t) = 3t²−6t on [0, 3]
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) = 3t²−6t on [0, 3]
Distance traveled and displacement Suppose an object moves along a line with velocity (in ft/s) v(t)=6−2t, for 0≤t≤6, where t is measured in seconds.
c. Find the distance traveled by the object on the interval 0≤t≤6.