Question

Problem 4 10 marks Two solid spheres of total masses m, and m,2 respectively collide such that at the instant of impact the x axis passes through the centers C1 and C2 of the spheres. The collision is perfectly elastic (that is, (vi - v2) =-(v^-v2) î). Find their velocities after impact (vi and v) in terms of their velocities before impact (vi and v2 or vi, V2,1 and 02). j C2 2/ 01

Answer 1

Since the collision is perfectly elastic,

Momentum before and after the collision is conserved,
Energy before and after the collision is conserved.

Applying momentum conservation,

On x-direction,

Before the collision,
The Momentum of both masses (system) = m₁v₁Cos( θ₁ ) + m₂v₂Cos( θ₂ )

After the collision,
The momentum of both masses (system) = m₁v'₁Cos( ϕ₁ ) + m₂v'₂Cos( ϕ₂ )

We get on conservation of momentum,
m₁v₁Cos( θ₁ ) + m₂v₂Cos( θ₂ ) = m₁v'₁Cos( ϕ₁ ) + m₂v'₂Cos( ϕ₂ ) ... 1st

On y-direction,

Before the collision,
The Momentum of both masses (system) = m₁v₁Sin( θ₁ ) + m₂v₂Sin( θ₂ )

After the collision,
The momentum of both masses (system) = m₁v'₁Sin( ϕ₁ ) + m₂v'₂Sin( ϕ₂ )

We get on conservation of momentum,
m₁v₁Sin( θ₁ ) + m₂v₂Sin( θ₂ ) = m₁v'₁Sin( ϕ₁ ) + m₂v'₂Sin( ϕ₂ ) ... 2nd

So, we have a system of 2 equations and 2 variables v'₁ and v'_{2 ,} on solving we get

v'₂ =  

[m\u1Cos(01) + m2U2Cos(02)Sin(01) - [m101 Sin(01) + m202Sin(02)Cos(01) [m2Cos(02) Sin(01) – m2Sin(02) Cos(01)]

v'_{1 =
[m\u1Cos(01) + m2U2Cos(02)]Sin(02) – [m101 Sin(01) + m202Sin(02)Cos(02) [m Cos(01) Sin(02) - m Sin(01) Cos(02)]}