Conservation of energy：練習問題

A ball of mass \(0.50\text{ kg}\) is dropped from a height of \(12.0\text{ m}\) above the ground. Due to air resistance, \(14.0\text{ J}\) of energy is dissipated as heat before it hits the ground. What is the speed of the ball just before impact? Take \(g = 9.81\text{ m s}^{-2}\).

A hydroelectric system uses water falling through a vertical distance of \(25\text{ m}\) at a rate of \(150\text{ kg s}^{-1}\). The water leaves the turbine with a speed of \(12\text{ m s}^{-1}\). If the overall efficiency of the energy conversion from water to electricity is \(60\%\), what is the electrical power output? Take \(g = 9.81\text{ m s}^{-2}\).

A spring with a stiffness constant of \(250\text{ N m}^{-1}\) is compressed by \(8.0\text{ cm}\) on a horizontal, frictionless surface. It is used to launch a toy car of mass \(0.20\text{ kg}\). What is the maximum speed the toy car can reach when the spring is released?

A trolley of mass \(5.0\text{ kg}\) moving at \(4.0\text{ m s}^{-1}\) collides with a stationary trolley of mass \(3.0\text{ kg}\). The two trolleys couple together and immediately begin to compress a horizontal spring of spring constant \(k = 200\text{ N m}^{-1}\). Calculate the maximum compression of the spring, assuming no external resistive forces act on the system.

A block of mass \(2.5\text{ kg}\) starts from rest and slides down a rough ramp of height \(4.0\text{ m}\). The ramp is inclined at an angle of \(30^\circ\) to the horizontal. If the block reaches the bottom with a speed of \(6.0\text{ m s}^{-1}\), what is the average frictional force acting on the block during its descent? Take \(g = 9.81\text{ m s}^{-2}\).

A block of mass \(0.50 \text{ kg}\) is released from rest at the top of a smooth slope inclined at \(40^\circ\) to the horizontal. Using the principle of conservation of energy, determine the speed of the block after it has traveled \(2.0 \text{ m}\) down the slope.

A spring with a spring constant of \(350 \text{ N m}^{-1}\) is compressed by \(0.12 \text{ m}\). This energy is used to launch a small projectile of mass \(0.040 \text{ kg}\) vertically upwards. Calculate the maximum height reached by the projectile from the point of release, assuming no energy losses.

A crate of mass \(50 \text{ kg}\) slides down a rough ramp that is \(6.0 \text{ m}\) long and inclined at \(25^\circ\) to the horizontal. If the crate starts from rest and reaches the bottom with a speed of \(4.5 \text{ m s}^{-1}\), calculate the work done against friction during the slide.

A small bead of mass \(0.050\text{ kg}\) is released from rest at point A and slides along a smooth curved wire to point B, as shown in the diagram. Point A is at a vertical height of \(0.80\text{ m}\) above point B.

(a) Calculate the theoretical speed of the bead at B assuming no friction.
(b) In a real experiment, the speed at B is measured to be \(3.5\text{ m s}^{-1}\). Calculate the work done against resistive forces during the motion.

A block of mass \(M = 2.0\text{ kg}\) is attached to a horizontal spring with spring constant \(k = 500\text{ N m}^{-1}\) on a frictionless surface. A second block of mass \(m = 0.50\text{ kg}\) moving at \(10\text{ m s}^{-1}\) collides with and sticks to block \(M\).

(a) Calculate the velocity of the combined blocks immediately after the collision.
(b) Calculate the maximum compression of the spring as the combined blocks come to rest momentarily.