Find the speed and direction of each ball considering collision is perfectly elastic.

Physics
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A ball of mass 2.5 kg moving with a speed of 4 m/s collides head on with a 1.5 ball at rest. if the collision is perfectly elastic, what will the speed and direction of each ball after the collision?

May 1st, 2015

This is a collision problem which is perfectly elastic.  In all collisions, total momentum is conserved.  When the collision is perfectly elastic, then total kinetic energy is conserved as well.  So start with momentum conservation:

p = mv

p_before = p_after            (where "p" stands for momentum)

(2.5 kg)(4 m/s) + (1.5 kg)(0 m/s)  =  (2.5 kg) * v1  +  (1.5kg) * v2                   (v1 is final velocity of 2.5kg object)

  (v2 is final velocity of 1.5kg object)

We have two unknown variables, so we need another equation.  Solve for one variable in terms of the other:

10 = 2.5*v1 + 1.5*v2

2.5*v1 = 10 - 1.5*v2

v1 = (10 - 1.5*v2) / 2.5

v1 = 4 - 0.6*v2

Now we need another equation.  Use the conservation of kinetic energy equation:

KE = 1/2 mv^2               (KE stands for kinetic energy)

KE_before  =  KE_after
1/2 (2.5kg)(4m/s)^2 + 1/2 (1.5kg)(0m/s)^2  =  1/2 (2.5kg)*v1^2  +  1/2 (1.5kg)*v2^2

20 = 1.25*v1^2  +  0.75*v2^2

Now, substitute the expression for "v1" from above into the kinetic energy equation:

20 = 1.25*(4 - 0.6*v2)^2  +  0.75*v2^2

20 = 1.25*(16 - 4.8*v2 + 0.36*v2^2)  +  0.75*v2^2

20 = 20 - 6*v2 + 0.45*v2^2 + 0.75*v2^2

20 = 20 - 6*v2 + 1.2*v2^2

1.2*v2^2 - 6*v2 = 0

v2*(1.2*v2 - 6) = 0

So we can get two solutions.

v2 = 0 m/s

Or,

1.2*v2 - 6 = 0

v2 = 6/1.2 = 5 m/s

We can eliminate v2 = 0 m/s answer because the 1.5 kg object started at rest and after the collision it would not remain at rest.  So, its final velocity must be v2 = 5 m/s

Now, to find the final velocity of the 2.5 kg object, substitute the velocity we just found into equation we found before:

v1 = 4 - 0.6*v2

v1 = 4 - 0.6*(5m/s)

v1 = 4 - 3 = 1 m/s

So, the velocity of the 2.5 kg object after the collision is 1 m/s in the same direction that it was moving in before the collision, and the velocity of the 1.5 kg object after the collision is 5 m/s in the same direction as the 2.5 kg object.

I hope that helps!

May 1st, 2015

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