第8章位置エネルギーとエネルギー保存

A boy throws a ball of mass $0.25\text{kg}$ straight upward with an initial speed of $20\text{m}\text{/}\text{s}$ When the ball returns to the boy, its speed is $17\text{m}\text{/}\text{s}$ How much much work does air resistance do on the ball during its flight?

A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance?

Using energy considerations and assuming negligible air resistance, show that a rock thrown from a bridge 20.0 m above water with an initial speed of 15.0 m/s strikes the water with a speed of 24.8 m/s independent of the direction thrown. (Hint: show that $K_{\text{i}}+U_{\text{i}}=K_{\text{f}}+U_{\text{f}})$

A 1.0-kg ball at the end of a 2.0-m string swings in a vertical plane. At its lowest point the ball is moving with a speed of 10 m/s. (a) What is its speed at the top of its path? (b) What is the tension in the string when the ball is at the bottom and at the top of its path?

Ignoring details associated with friction, extra forces exerted by arm and leg muscles, and other factors, we can consider a pole vault as the conversion of an athlete’s running kinetic energy to gravitational potential energy. If an athlete is to lift their body 4.8 m during a vault, what speed must they have when they plant the pole?

Tarzan grabs a vine hanging vertically from a tall tree when he is running at $9.0\text{m}\text{/}\text{s}.$ (a) How high can he swing upward? (b) Does the length of the vine affect this height?

Assume that the force of a bow on an arrow behaves like the spring force. In aiming the arrow, an archer pulls the bow back 50 cm and holds it in position with a force of $150\text{N}$. If the mass of the arrow is $50\text{g}$ and the “spring” is massless, what is the speed of the arrow immediately after it leaves the bow?

A $100-\text{kg}$ man is skiing across level ground at a speed of $8.0\text{m/s}$ when he comes to the small slope 1.8 m higher than ground level shown in the following figure. (a) If the skier coasts up the hill, what is his speed when he reaches the top plateau? Assume friction between the snow and skis is negligible. (b) What is his speed when he reaches the upper level if an $80-\text{N}$ frictional force acts on the skis?

A sled of mass 70 kg starts from rest and slides down a $10\text{°}$ incline $80\text{m}$ long. It then travels for 20 m horizontally before starting back up an $8\text{°}$ incline. It travels 80 m along this incline before coming to rest. What is the magnitude of the net work done on the sled by friction?

A girl on a skateboard (total mass of 40 kg) is moving at a speed of 10 m/s at the bottom of a long ramp. The ramp is inclined at $20\text{°}$ with respect to the horizontal. If she travels 14.2 m upward along the ramp before stopping, what is the net frictional force on her?

A baseball of mass 0.25 kg is hit at home plate with a speed of 40 m/s. When it lands in a seat in the left-field bleachers a horizontal distance 120 m from home plate, it is moving at 30 m/s. If the ball lands 20 m above the spot where it was hit, how much work is done on it by air resistance?

A small block of mass m slides without friction around the loop-the-loop apparatus shown below. (a) If the block starts from rest at A, what is its speed at B? (b) What is the force of the track on the block at B?

The massless spring of a spring gun has a force constant $k=12\text{N/m}.$ When the gun is aimed vertically, a 15-g projectile is shot to a height of 0.50 m above the end of the expanded spring. (See below.) How much was the spring compressed initially?

A small ball is tied to a string and set rotating with negligible friction in a vertical circle. If the ball moves over the top of the circle at its slowest possible speed (so that the tension in the string is negligible), what is the tension in the string at the bottom of the circle, assuming there is no additional energy added to the ball during rotation?