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## Homework Statement

Consider an elastic collision between two bodies of equal mass, one of which is initially at rest. Let their velocities be [itex]\vec{v_1}[/itex] and [itex]\vec{v_2} = \vec{0}[/itex] before the collision, and [itex]\vec{v_1 '}[/itex] and [itex]\vec{v_2 '}[/itex] after the collision. Write down the vector equation representing conservation of momentum and the scalar equation which expresses that the collision is elastic. Use these to prove that [itex]\vec{v_1 '}[/itex] and [itex]\vec{v_2 '}[/itex] are orthogonal.

## Homework Equations

## The Attempt at a Solution

If the collision is elastic, then KE = KE'.

m1 = m2 = m

[itex]\vec{p} = \vec{p'}[/itex]

If the two velocity vectors are orthogonal, then [itex]\vec{v_1 '} \cdot \vec{v_2 '} = 0[/itex].

Conservation of energy:

[itex]\frac{1}{2} mv_1^2 = \frac{1}{2} m v_1^{'2} + \frac{1}{2} m v_2^{'2}[/itex]

[itex]v_1^2 = v_1^{'2} + v_2^{'2}[/itex], which can be written as [itex]| \vec{v_1} |^2 =| \vec{v_1 '} |^2 + | \vec{v_2 '} |^2[/itex], although I am not sure of how helpful this will be.

Conservation of momentum:

[itex]m \vec{v_1} + \vec{0} = m \vec{v_1 '} + m \vec{v_2 '}[/itex]

[itex]\vec{v_1} = \vec{v_1 '} + \vec{v_2 '}[/itex]

Taking the magnitude, [itex]| \vec{v_1} | = | \vec{v_1 '} |+ | \vec{v_2 '} |[/itex]; again, I am not certain of how helpful this will be.

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Up to this point, I am not sure as to how to proceed. Could someone provide me with a hint?