Answer to Question #84220 in Classical Mechanics for Dave

Question

A 1.50 kg wooden block is hanging motionless from a 2.00 m massless string. A 125 g arrow is
shot at 55.0 m/s at the block and sticks into it, causing the block to swing.
- What is the maximum height the block/arrow system will swing with respect to the
original position of the block? (0.912 m)
- What would be different about this situation if the arrow passed through the block, but
slowed down in the process? Explain in terms of momentum and energy

Answer 1

The law of the conservation of the momentum gives

mv_i=(m+M) v_f

The law of the conservation of the energy gives

((m+M) v_f^2)/2=(m+M)gH

So, the maximum height the block/arrow system

H=(v_f^2)/2g=1/2g [(mv_i)/(m+M)]^2=1/(2×9.81) [(0.125×55.0)/(0.125+1.50)]^2=0.912 m

If the arrow passed through the block, then the maximum height the block decreases. This is because law of the conservation of the momentum in this case gives

mv_i=mv_a+Mv_b

So, the initial speed of the block is less with respect to previous case

v_b<v_f=(mv_i)/(m+M)