Proving magnitude of impulse on either spheres

[toforfiltum]

Homework Statement

A sphere of mass m is moving with a speed V along a horizontal straight line. It collides with an identical sphere of mass m moving along the same straight line with speed u (u<V). Show that the magnitude of impulse on either sphere is
½m(1+e)(V-u), where e is the coefficient of restitution between the two spheres.

Homework Equations

m₁u₁ +m₂u₂ = m₁v₁ +m₂v₂
V₂-V₁/ u₁-u₂= e

The Attempt at a Solution

I calculated impulse on sphere with velocity V. I =m(V₁-V). Using coefficient of restitution, I get I = m(u₁-e(V-u)-V). Using equation formed from conservation of momentum, I get I =m[u-V₁- e(V-u)]. I have trouble getting rid of the u and V₁ terms. How do I go about this?

 

[mfb]

Your notation is confusing. How are V₁ and so on related to the V and u given in the problem statement?

 

[toforfiltum]

Your notation is confusing. How are V₁ and so on related to the V and u given in the problem statement?

V₁ and u₁ are the velocities after collision.

 

[mfb]

What are V₂ and u₂ then?

 

[toforfiltum]

What are V₂ and u₂ then?

Oh, that's just the general formula.

 

[haruspex]

V₁ and u₁ are the velocities after collision.

I think you mean those are the velocities of the first ball, after and before collision respectively. In the context of the question, u₂ = u, and u₁ = V.
Your two 'relevant equations' have two unknowns, the two velocities after impact. Use them to express those velocities in terms of the given velocities and e.

 

[toforfiltum]

I think you mean those are the velocities of the first ball, after and before collision respectively. In the context of the question, u₂ = u, and u₁ = V.
Your two 'relevant equations' have two unknowns, the two velocities after impact. Use them to express those velocities in terms of the given velocities and e.

Okay, to be clear, I define V and u as the initial speeds of the balls, and V₁ and u₁ as the speeds after collision.
So, since impulse is the same for both balls but in opposite direction, I form an equation m(V₁- V) = -m(u₁-u). Therefore I get m(V₁ + u₁) =m(V =u). However, coefficient of restitution gives me u₁-v₁ = e(V-u), and the u₁-V₁ term is not the same as in the first equation, so I can't make a complete substitution. How do I go from here?

 

[mfb]

You can solve the last equation for u₁ and plug it into the first one. That allows to solve for V₁ which is the last unknown in the system. Then you can calculate the magnitude of impulse.