Momentum：練習問題

A particle P of mass \(0.5 \text{ kg}\) is moving with a speed of \(8 \text{ m s}^{-1}\) on a smooth horizontal surface. It strikes a stationary particle Q of mass \(1.5 \text{ kg}\). The two particles coalesce on impact. Find the common speed of the particles after the impact.

A stationary explosive shell of mass \(5 \text{ kg}\) explodes into two fragments of mass \(2 \text{ kg}\) and \(3 \text{ kg}\). The \(2 \text{ kg}\) fragment is projected horizontally with a speed of \(15 \text{ m s}^{-1}\). Find the speed of the \(3 \text{ kg}\) fragment.

A particle A of mass \(m\) is moving with speed \(v\) on a smooth horizontal surface when it strikes a stationary particle B of mass \(km\), where \(k > 1\). After the impact, the particles move in opposite directions, and the total kinetic energy of the system is reduced by \(25\%\). If particle A has a speed of \(v_A\) and particle B has a speed of \(v_B\) after the impact, find the value of the constant \(k\) such that \(v_A = v_B\).

Particle P of mass \(2m\) is moving with speed \(3u\) on a smooth horizontal plane and collides with particle Q of mass \(m\) which is moving in the same direction with speed \(u\). The two particles coalesce on impact. Find the ratio of the speed of the combined particle after impact to the initial speed of Q.

A sphere A of mass \(2 \text{ kg}\) is moving on a smooth horizontal floor with a velocity of \(6 \text{ m s}^{-1}\). It hits a sphere B of mass \(4 \text{ kg}\) which is moving towards A with a velocity of \(3 \text{ m s}^{-1}\). After the impact, sphere A rebounds with a speed of \(2 \text{ m s}^{-1}\). Calculate the speed of sphere B immediately after the impact.

A particle P of mass \(m\) is moving with speed \(4u\) on a smooth horizontal plane and collides with a particle Q of mass \(km\) (where \(k\) is a constant) which is moving in the same direction with speed \(u\). After the collision, P is reduced to rest and Q continues with speed \(2u\).

(a) Show that \(k = 3\).
(b) Given that the loss in total kinetic energy due to the impact is \(1.5 \text{ J}\) and \(u = 2 \text{ m s}^{-1}\), find the value of \(m\).

A particle A of mass \(0.2 \text{ kg}\) is released from rest at the top of a smooth plane inclined at an angle \(30^\circ\) to the horizontal. The length of the plane is \(2.5 \text{ m}\). At the bottom of the plane, it hits a stationary particle B of mass \(0.3 \text{ kg}\) on a smooth horizontal surface. The particles coalesce on impact.

(a) Calculate the speed of A just before it reaches the bottom of the plane.
(b) Find the speed of the combined particle immediately after the impact.
(c) Calculate the loss in kinetic energy caused by the impact.