Answer to Question #116101 in Mechanics | Relativity for Sabbrina

Question

A dart with mass md is launched toward a block of mass mb that is suspended from a string of length L, as shown at left above. The dart is moving horizontally with speed b immediately before it strikes the block and remains embedded. The dart-block system then swings up to a point at which it’s center of mass reaches a maximum height H above it’s starting position, as shown at the right above. The blocks mass md is greater than the darts mass md. A) Indicate which object, the dart or the block, if either, experience an impulse of larger magnitude during the collision. If the impulse is the same magnitude for both objects, state this explicitly, Briefly explain your reasoning. B) If the speed of the dart as it embeds itself into the block is greater than v, how would the maximum height reached by the center of mass of the dart-block system compare to H? Explain your response without deriving or manipulating equations.

Answer 1

A) Since the dart hits the hanging mass and sticks to it, the time of interaction is equal for both bodies. According to Newton's second law, the force caused by the dart on the block is the same as the force caused by block on the dart. Therefore, the impulse is equal for both objects.

B) The block and the dart would reach higher height. According to energy conservation, calculate the height H:

$$\frac{1}{2}m_du^2=(m_d+m_b)gH,\\ \space\\ H=\frac{m_du^2}{g(m_d+m_b)}.$$

We see that the height is directly proportional to the speed of the dart. Therefore, higher speed would cause the block and the dart go higher.