(III) An object moving vertically has ${\overrightarrow{v}}_{}={\overrightarrow{v}}_0$ at t = 0. Determine a formula for its velocity as a function of time assuming a resistive force F = -bv as well as gravity for two cases: ${\overrightarrow{v}}_0$ is upward.

Two rock climbers, Paul and Jeanne, use safety ropes of similar length. Jeanne's rope is more elastic, called a dynamic rope by climbers. Paul has a static rope, not recommended for safety reasons. In a similar fall, Paul's rope stretches by only 30 cm. How many times his weight will the rope pull on him? Which climber is more likely to be hurt?

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You are driving home in your 860-kg car at 15 m/s. At a point 45 m from the beginning of an intersection, you see a green traffic light change to yellow, which you expect will last 4.0 s, and the distance to the far side of the intersection is 65 m (Fig. 4–71). If you choose to accelerate, your car's engine will furnish a forward force of 1200 N. Will you make it completely through the intersection before the light turns red?

Suppose a 65-kg person jumps from a height of 3.0 m down to the ground. Estimate the decelerating force if the person lands stiff-legged so d ≈ 1.0 cm.

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(II) The drag force on large objects such as cars, planes, and sky divers moving through air is more nearly F_D = -cv². For this quadratic dependence on v, determine a formula for the terminal velocity v_T, of a vertically falling object..

Air resistance acting on a falling body can be taken into account by the approximate relation for the acceleration: a = dv/dt = g ― kv, where k is a constant. Determine an expression for the terminal velocity, which is the maximum value the velocity reaches.

(II) A 2.0-kg purse is dropped from the top of the Leaning Tower of Pisa and falls 55 m before reaching the ground with a speed of 27m/s. What was the average force of air resistance?

(II) The terminal velocity of a 3 x 10^-5 kg raindrop is about 9 m/s. Assuming a drag force F_D = -bv, determine the time required for such a drop, starting from rest, to reach 63% of terminal velocity.