6. Intro to Forces (Dynamics)

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. Jeanne (Fig. 4–72) falls freely about 2.0 m and then the rope stops her over a distance of 1.0 m. Estimate how large a force (assume constant) she will feel from the rope. (Express the result in multiples of her weight.)

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A 25-g projectile is fired into a cube of ballistic gel at a velocity of 360 m/s. If the projectile penetrates 15 cm into the gel before stopping, find the average force exerted by the gel onto the projectile. Use kinematics and dynamics (Newton's laws).

(III) A particle of mass m, initially at rest at x = 0, is accelerated by a force that increases in time as F = Ct². Determine its velocity v and position x as a function of time.

At the instant a race began, a 65-kg sprinter exerted a force of 720 N on the starting block at a 22° angle with respect to the ground. What was the horizontal acceleration of the sprinter?

At the instant a race began, a 65-kg sprinter exerted a force of 720 N on the starting block at a 22° angle with respect to the ground. If the force was exerted for 0.32 s, with what speed did the sprinter leave the starting block?

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..

(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.

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.