# GRE GR 1777, Problem 002 Solution

Physics GRE GR 1777

2. Classical Mechanics (Momentum and Energy)

Solution)

By the linear momentum conservation,

1) x-direction : $m_{1}v_{1,x} + m_{2}v_{2,x} = (m_{1} + m_{2})v_{x}$

$(0.5) \cdot 2 + (1.0)\cdot 0 = (0.5 + 1.5) v_{x}$
$\therefore v_{x} = \frac{2}{3}$

2) y-direction : $m_{1}v_{1,y} + m_{2}v_{2,y} = (m_{1} + m_{2})v_{y}$

$(0.5) \cdot 0 + (1.0) \cdot 1 = (0.5 + 1.5) v_{y}$
$\therefore v_{y} = \frac{2}{3}$

Therefore, $v = \frac{2\sqrt{2}}{3}$

Also, using work-energy theorem,

$W = \Delta K.E = \Delta \frac{1}{2}mv^2$
$W = \frac{1}{2}((0.5) \cdot 2^2 + 1 \cdot 1^2) - \frac{1}{2}((1.5) \cdot (\frac{2\sqrt{2}}{3})^2) = \frac{5}{6} J$

If we assume that all energy generated by collision is converted to heat, then the heat energy produced by the collision is $\frac{5}{6} J$