Engineering Mechanics

(^564) „„„„„ A Textbook of Engineering Mechanics
The Newton’s Law of Collision of Elastic Bodies, also holds good for indirect impact i.e.,
(v 2 cos θ 2 – v 1 cos θ) = e (u 1 cos α 1 – u 2 cos α 2 )
Example 27.7. A ball of mass 2 kg, moving with a velocity of 3 m/sec, impinges on a ball of
mass 4 kg moving with a velocity of 1 m/sec. The velocities of two balls are parallel and inclined at
30° to the line joining their centres at the instant of impact. If the coefficient of restitution be 0.5, find
(a) direction, in which the 4 kg ball will move after impact ;
(b) velocity of the 4 kg ball after impact ;
(c) direction, in which the 2 kg ball will move after impact ; and
(d) velocity of the 2 kg ball after impact.
Solution. Given : Mass of first ball (m 1 ) = 2 kg ; Initial velocity of first ball (u 1 ) = 3 m/s ;
Mass of second ball (m 2 ) = 4 kg ; Initial velocity of second ball (u 2 ) = 1 m/s ; Angle, which initial
velocity of first ball makes with the line of impact (α 1 ) = 30° ; Angle, which initial velocity of second
ball makes with the line of impact (α 2 ) = 30° and coefficient of restitution (e) = 0.5
Fig. 27.4.
(a) Direction, in which the 4 kg ball will move after the impact
Let θ 1 = Angle, which the 2 kg ball makes with the line of impact,
θ 2 = Angle, which the 4 kg ball makes with the line of impact,
v 1 = Velocity of the 2 kg ball after impact, and
v 2 = Velocity of the 4 kg ball after impact,
We know that the components of velocities, perpendicular to the line of impact, remain
unchanged before and after impact.
∴ v 1 sin θ 1 = u 1 sin α 1 = 3 sin 30° = 3 × 0.5
or v 1 sin θ 1 = 1.5 ...(i)
Similarly v 2 sin θ 2 = u 2 sin α 2 = 1 sin 30° = 1 × 0.5
or v 2 sin θ 2 = 0.5 ...(ii)
We also know from the law of conservation of momentum
m 1 u 1 cos α 1 + m 2 u 2 cos α 2 = m 1 v 1 cos θ 1 + m 2 v 2 cos θ 2
(2 × 3 cos 30°) + (4 × 1 × cos 30°) = 2v 1 cos θ 1 + 4v 2 cos θ 2
(6 × 0.866) + (4 × 0.866) = 2v 1 cos θ 1 + 4v 2 cos θ 2
8.66 = 2v 1 cos θ 1 + 4v 2 cos θ 2
∴ v 1 cos θ 1 + 2v 2 cos θ 2 = 4.33 ...(iii)