Question

A 1kg ball moving at 12m/s collides head on with 2kg ball moving in opposite direction at 24m/s.The velocity of each ball after impact, if the coefficient of restitution is 2/3is

Answer 1

Answer: -28 m/s and -4 m/s

Explanation:

From law of conservation of momentum, when there is no net external force is acting on the system, the momentum is conserved.

This means, Momentum before the collision = Momentum after the collision

⇒m₁u₁+m₂u₂=m₁v₁+m₂v₂

where, m₁ and m₂ are the masses of the colliding objects, u₁ and u₂ are their initial velocities and v₁ and v₂ are their final velocities.

It is given that:

m₁ = 1 kg

m₂ = 2 kg

u₁ = 12 m/s

u₂ = -24 m/s

The coefficient of restitution,

$e =\frac{v_2-v_1}{u_1-u_2}$

$e = \frac {2}{3}$

⇒2/3 (12-(-24))=v₂-v₁

⇒24 =v₂-v₁

⇒v₂=24+v₁ ... (1)

From law of conservation of momentum,

1 kg × 12 m/s + 2 kg × -24 m/s = 1 kg v₁+ 2 kg v₂

⇒12-48 =v₁+2v₂ ⇒ v₁+2v₂= -36... (2)

Substitute (1) into (2)

⇒v₁+2×(24+v₁)=-36

⇒3v₁+48=-36

⇒3v₁=-84 ⇒v₁=-28 m/s

Substitute this value in (1)

v₂= 24+(-28) =-4 m/s

Hence, the velocity of ball of mass 1 kg after collision is -28 m/s and ball with mass 2 kg, - 4m/s. This means, both the balls move in same direction after collision.

Answer 2

m1 u1 + m2 u2 = m1 v1 + m2 v2 …............. [1]  

where:  

m1 = 1 kg  

m2 = 2 kg  

u1 = 12 m/s  

u2 = -24 m/s  

and the coefficient of restitution, e, is given by:  

e = (v2 – v1)/(u1 – u2) …......................... [2]  

 a) e = 2/3  

[1] …............... 1 * 12 + 2 * (-24) = 1 * v1 + 2 * v2  

v1 + 2 v2 = -36  

v1 = -2 v2 - 36  

[2] …................ 2/3 = (v2 – v1)/36  

2/3 = (v2 – (-2 v2 – 36))/36  

72/3 = 3 v2 + 36  

► v2 = -12/3 = -4 m/s  

► v1 = -2 v2 – 36 = -28 m/s  

b)  

[1] …............... m1 u1 + m2 u2 = (m1 + m2) v  

1 * 12 + 2 * (-24) = (1 + 2) v  

3 v = -36  

 v = -12 m/s  

c) e = 1  

[1] …............... 1 * 12 + 2 * (-24) = 1 * v1 + 2 * v2  

v1 + 2 v2 = -36  

v1 = -2 v2 - 36  

[2] …................ 1 = (v2 – v1)/36  

1 = (v2 – (-2 v2 – 36))/36  

36 = 3 v2 + 36  

► v2 = 0  

► v1 = -2 v2 – 36 = -36 m/s  

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