To hoist himself into a tree, a 72.0-kg man ties one end of a nylon rope around his waist and throws the other end over a branch of the tree. He then pulls downward on the free end of the rope with a force of 358 N. Neglect any friction between the rope and the branch, and determine the man's upward acceleration.

A 292-kg motorcycle is accelerating up along a ramp that is inclined $30.0^{\circ}$ above the horizontal. The propulsion force pushing the motorcycle up the ramp is 3150 N, and air resistance produces a force of 250 N that opposes the motion . Find the magnitude of the motorcycle's acceleration.

A penguin slides at a constant velocity of 1.4 m/s down an icy incline. The incline slopes above the horizontal at an angle of

$$6.9^\circ.$$

At the bottom of the incline, the penguin slides onto a horizontal patch of ice. The coefficient of kinetic friction between the penguin and the ice is the same for the incline as for the horizontal patch. How much time is required for the penguin to slide to a halt after entering the horizontal patch of ice?

A small sphere is hung by a string from the ceiling of a van. When the van is stationary, the sphere hangs vertically. However, when the van accelerates, the sphere swings backward so that the string makes an angle of

$$\theta$$

with respect to the vertical. (a) Derive an expression for the magnitude a of the acceleration of the van in terms of the angle

$$\theta$$

and the magnitude g of the acceleration due to gravity. (b) Find the acceleration of the van when

$$\theta = 10.0 ^ { \circ }$$

(c) What is the angle

$$\theta$$

when the van moves with a constant velocity?

A car is towing a boat on a trailer. The driver starts from rest and accelerates to a velocity of + 11 m/s in a time of 28 s. The combined mass of the boat and trailer is 410 kg. The frictional force acting on the trailer can be ignored. What is the tension in the hitch that connects the trailer to the car?

A 15-g bullet is fired from a rifle. It takes

$$2.50 \times 10 ^ { - 3 } \mathrm { s }$$

for the bullet to travel the length of the barrel, and it exits the barrel with a speed of 715 m/s. Assuming that the acceleration of the bullet is constant, find the average net force exerted on the bullet.