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Rotational Work and Energy

A particle of...

Question

A particle of mass $m_{1}$ and speed $v_{1}$ (in the +x direction) collides with another particle of mass $m_{2}$ is at rest before the collision occurs, thus $v_{2} = 0$ . After the collision, the particles have velocities $v_{1}^{'}$ and $v_{2}^{'}$ in the x-y plane in the directions of $θ_{1}$ and $θ_{2}$ with the x-axis (see figure). There re two external forces. Express all out answers in terms of $m_{1}, m_{2}, v_{1}, θ_{1}$ and $θ_{2}$
(a) What is the total momentum before the collision (direction and magnitude)
(b) What is the total momentum after the collision (direction and magnitude)
(c) What is the total kinetic energy before the collision?
(d) What is the ratio of the speed $\frac{v_{2}^{'}}{v_{1}^{'}}$
(e) What is the magnitude (speed) of $v_{1}^{'}$ , if $m_{1} = m_{2}$ ?

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Solution

$(a) .$ Momentum before collision $= m_{1} v_{1}$ $^i$
$(b) .$ Momentum is conserb after and before collision . so momentum after collision $= m_{1} v_{1}^i$
$(c) .$ Total K.E. before collision $= \frac{1}{2} m_{1} {v_{1}}^{2}$
$(d) .$ By momentum conservation.
$\Rightarrow$ $m_{1} v_{1} cos θ_{2} = m_{2} {v_{2}}^{1}$
${v_{2}}^{1} = \frac{m_{1}}{m_{2}} v_{1} cos θ_{2}$
Similarly,
$\Rightarrow m_{1} v_{1} cos θ_{1} = m_{1} {v_{1}}^{1}$
${v_{1}}^{1} = v_{1} cos θ_{1}$
$\Rightarrow$ Ratio $\frac{{v_{2}}^{1}}{{v_{1}}^{1}} = \frac{m_{1}}{m_{2}} \frac{cos θ_{2}}{cos θ_{1}}$
$(e) . {v_{1}}^{1} = v_{1} cos θ_{1}$
Hence, solved.

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