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.
Answer/Explanation
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.
Answer/Explanation
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. [2]
(b) Find the loss of kinetic energy of the system due to the collision. [3]
Answer/Explanation
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.
Answer/Explanation
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 = \(1 ms^{-1}\)