Solving Elastic Collision - Get Help Now!

[johnhuntsman]

A cart with a mass of 340 g moving on a frictionless linear air
track at an initial speed of 1.2 m/s undergoes an elastic collision with an
initially stationary cart of unknown mass. After the collision, the rst cart
continues in the same direction at 0.66 m/s. (a) What is the mass of the
second cart? (b) What is its speed after the collision?

I have two equations: Conservation of Momentum and Conservation of KE (since this is elastic).

m₁v_i = m₁v₁ + m₂v₂

.5m₁v_i² = .5m₁v₁² + .5m₂v₂²

I've solved for v₂ using the equation for Conservation of Momentum, and plugged that into the equation for Conservation of KE and my answer is wrong (used up about a page of algebra, please don't expect me to type it out all on here). The correct answer can be found using the method in the PDF below.

http://www.physics.ucc.ie/py1052_ps6.pdf (problem 8)

My issue is that they solve for v_1f (or as I called it in the equations above, v₁) and yet v_2f (or v₂) is nowhere to be found on the side opposite of their v_1f. I don't see how their algebra makes any sense. All I want is someone to explain this to me as this is supposed to be a straightforward question.

 

[haruspex]

.5m₁v_i² = .5m₁v₁² + m₂v₂²

You missed a 0.5 there. Is it as simple as that?

My issue is that they solve for v_1f (or as I called it in the equations above, v₁) and yet v_2f (or v₂) is nowhere to be found on the side opposite of their v_1f.

They used the momentum equation to get v_2f in terms of the other variables, then substituted that in the KE equation.

 

[johnhuntsman]

They used the momentum equation to get v_2f in terms of the other variables, then substituted that in the KE equation.

v_2f = [m₁v_i - m₁v_f] / m₂

Substitute that into cons. of KE:

.5m₁v_i² = .5m₁v₁² + .5m₂[(m₁v_i - m₁v₁) / m₂]²

Which still doesn't look like what they have. This is what I solved and I ended up getting .127 kg. What mistake have I made at this point that I'm not seeing?

P.S. Eqn. for conservation of KE is fixed.

 

[haruspex]

.5m₁v_i² = .5m₁v₁² + .5m₂[(m₁v_i - m₁v₁) / m₂]²

Which still doesn't look like what they have.

Work with that some. You should find v_i - v₁ cancels, getting rid of the quadratic.

 

[TamuNinja]

use this after you get the numbers from momentum equation of:

ƩP_i=ƩP_f

(you have the break down of the above correct?)Now plug into this

0=ΔKE+ΔPEg+ΔPE_spring
=1/2mv²_f-1/2mv²_f+mgh_f-mgh_i+1/2kx²_f-1/2kx²_f

Now just place you values in the equation and remove things that you don't use for example the spring.

 

[johnhuntsman]

Work with that some. You should find v_i - v₁ cancels, getting rid of the quadratic.

Alrighty, thanks. That did it. I appreciate it : D