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
The answer is (E)

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