8.3 Conservation of Energy

How high above the bottom of its arc is the particle in the simple pendulum above, when its speed is $0.81 \text{m} / \text{s} ?$

You probably recall that, neglecting air resistance, if you throw a projectile straight up, the time it takes to reach its maximum height equals the time it takes to fall from the maximum height back to the starting height. Suppose you cannot neglect air resistance, as in the example. Is the time the projectile takes to go up (a) greater than, (b) less than, or (c) equal to the time it takes to c…

What potential energy $U \left(x\right)$ can you substitute in the equation that will result in motion with constant velocity of 2 m/s for a particle of mass 1 kg and mechanical energy 1 J?

“$E = K + U$ constant is a special case of the work-energy theorem.” Discuss this statement.

A child jumps up and down on a bed, reaching a higher height after each bounce. Explain how the child can increase his maximum gravitational potential energy with each bounce.

Neglecting air resistance, how much would I have to raise the vertical height if I wanted to double the impact speed of a falling object?

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

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 $K_{\text{i}} + U_{\text{i}} = K_{\text{f}} + U_{\text{f}} \left.\right)$

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?

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 sled of mass 70 kg starts from rest and slides down a $10 °$ incline $80 \text{m}$ long. It then travels for 20 m horizontally before starting back up an $8 °$ 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 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?