The momentum of a moving object:

$\overline{){\mathbf{p}}{\mathbf{=}}{\mathbf{m}}{\mathbf{v}}}$

Conservation of momentum:

$\overline{){{\mathbf{m}}}_{{\mathbf{1}}}{{\mathbf{v}}}_{\mathbf{0}\mathbf{,}\mathbf{1}}{\mathbf{}}{\mathbf{+}}{\mathbf{}}{{\mathbf{m}}}_{{\mathbf{2}}}{{\mathbf{v}}}_{\mathbf{0}\mathbf{,}\mathbf{2}}{\mathbf{}}{\mathbf{=}}{\mathbf{}}{{\mathbf{m}}}_{{\mathbf{1}}}{{\mathbf{v}}}_{\mathbf{f}\mathbf{,}\mathbf{1}}{\mathbf{}}{\mathbf{+}}{\mathbf{}}{{\mathbf{m}}}_{{\mathbf{2}}}{{\mathbf{v}}}_{\mathbf{f}\mathbf{,}\mathbf{2}}}$

Conservation of kinetic energy:

$\overline{)\frac{\mathbf{1}}{\mathbf{2}}{{\mathbf{m}}}_{{\mathbf{1}}}{{\mathbf{\left(}}{{\mathbf{v}}}_{\mathbf{0}\mathbf{,}\mathbf{1}}{\mathbf{\right)}}}^{{\mathbf{2}}}{\mathbf{}}{\mathbf{+}}{\mathbf{}}\frac{\mathbf{1}}{\mathbf{2}}{{\mathbf{m}}}_{{\mathbf{2}}}{{\mathbf{\left(}}{{\mathbf{v}}}_{\mathbf{0}\mathbf{,}\mathbf{2}}{\mathbf{\right)}}}^{{\mathbf{2}}}{\mathbf{}}{\mathbf{=}}{\mathbf{}}\frac{\mathbf{1}}{\mathbf{2}}{{\mathbf{m}}}_{{\mathbf{1}}}{{\mathbf{\left(}}{{\mathbf{v}}}_{\mathbf{f}\mathbf{,}\mathbf{1}}{\mathbf{\right)}}}^{{\mathbf{2}}}{\mathbf{}}{\mathbf{+}}{\mathbf{}}\frac{\mathbf{1}}{\mathbf{2}}{{\mathbf{m}}}_{{\mathbf{2}}}{{\mathbf{\left(}}{{\mathbf{v}}}_{\mathbf{f}\mathbf{,}\mathbf{2}}{\mathbf{\right)}}}^{{\mathbf{2}}}}$

**A. **

The objects separate after the collision.

Using conservation of momentum and the variables m_{1} = m_{2} = m, v_{0,1} = v, v_{0,2} = 0, v_{f,1} = v_{1} and v_{f,2} = v_{2}.

mv + m(0) = mv_{1} + mv_{2}

v_{1} = v - v_{2} --------- Eqn 1

Using conservation of energy:

(1/2)mv^{2} = (1/2)mv_{1}^{2} + (1/2)mv_{2}^{2}

v^{2} = v_{1}^{2} + v_{2}^{2} ------- Eqn 2

Let two particles of equal mass collide. Particle 1 has initial velocity, directed to the right, and particle 2 is initially stationary.

A. If the collision is elastic, what are the final velocities and of particles 1 and 2?

B. Now suppose that the collision is perfectly inelastic. What are the velocities and of the two particles after the collision?

B. Now assume that the mass of particle 1 is 2m, while the mass of particle 2 remains m. If the collision is elastic, what are the final velocities v_{1 }and v_{2} of particles 1 and 2?

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