# CIE A level Math -Mechanics : 4.3 Momentum:linear momentum: Exam Style Questions Paper 4

Two small smooth spheres A and B, of equal radii and of masses km kg and m kg respectively, where k > 1, are free to move on a smooth horizontal plane. A is moving towards B with speed 6 m s−1 and B is moving towards A with speed 2 m s−1. After the collision A and B coalesce and move with speed 4 m s−1.

### (a) Question

Find k.

Attempt at use of conservation of momentum

km × 6 – m × 2 =  (km + m)× 4

k = 3

### (b)Question

Find, in terms of m, the loss of kinetic energy due to the collision.

Ans:

KE initial = $$\frac{1}{2}\times km\times 6^{2} + \frac{1}{2}\times m\times (-2)^{2}$$

KE after = $$\frac{1}{2}\left ( km+m) \right )\times 4^{2}$$

Loss of KE = 24m J

### Question

A particle B of mass 5 kg is at rest on a smooth horizontal table. A particle A of mass 2.5 kg moves on the table with a speed of 6 m s−1 and collides directly with B. In the collision the two particles coalesce.
(a) Find the speed of the combined particle after the collision.                                                                                 

(b) Find the loss of kinetic energy of the system due to the collision.                                                                       

Ans

(a) 6 × 2.5 = 2.5 v + 5 v
v = 2 ms–1

(b) Use KE = ½ mv2 either before or after collision
KE(before) = 0.5 × 2.5 × 62
KE(after) = 0.5 × 7.5 × 22
Loss of KE = 30 J

### Question

Two particles P and Q of masses  kg and  kg respectively are free to move in a horizontal straight line on a smooth horizontal plane. P is projected towards Q with speed $$0.5ms^{-1}$$. At the same instant Q is projected towards P with speed $$1ms^{-1}$$. Q comes to rest in the resulting collision.

Find the speed of P after the collision.

Ans:

$$±0.2\times 0.5$$ or $$±0.3 \times 1$$
$$0.2 \times 0.5 + 0.3 \times (-1)=-0.2 \times v + 0$$
Speed = $$1 ms^{-1}$$

### Question A uniform solid cone has weight 5 N and base radius 0.1 m. AB is a diameter of the base of the cone.
The cone is held in equilibrium, with A in contact with a rough horizontal surface and AB vertical, by a force applied at B. This force has magnitude 3 N and acts parallel to the axis of the cone (see
diagram). Calculate the height of the cone.

Conservation of momentum at$$\frac{h}{4} \frac{5\times h}{4}=3\times 0.2$$ (h=)0.48m