A pendulum consists of a 2.0-kg block hanging on a 1.5-m length string. A 10-g bullet moving with a horizontal velocity of 900 m/s strikes, passes through, and emerges from the block (initially at rest) with a horizontal velocity of 300 m/s. To what maximum height above its initial position will the block swing?
A) 32 cm
B) 38 cm
C) 46 cm
D) 27 cm
E) 9 cm

Question

Quizplus · Accepted Answer

First, we need to find the initial momentum of the bullet-block system, which results in an equal and opposite final momentum after the collision. Let the initial velocity of the bullet be v, then:
Initial momentum = mv = (0.01 kg)(900 m/s) = 9 kg*m/s
Final momentum = (2 kg + 0.01 kg)(v') where v' is the common velocity of the bullet and the block after the collision.
Due to conservation of momentum, we have:
mv = (2.01 kg)(v')
v' = mv / (2.01 kg) = (0.01 kg)(900 m/s) / (2.01 kg) = 4.48 m/s
The kinetic energy of the system is not conserved, and some of it is lost due to the collision. However, conservation of energy tells us that the initial potential energy of the block-bullet system is equal to the sum of their final kinetic energies (after the bullet has left the block).
Initial potential energy = mgh = (2 kg)(9.81 m/s^2)(1.5 m) = 29.43 J
Final kinetic energy = (1/2)mv'^2 + (1/2)M(V')^2 where M is the mass of the block, and V' is its velocity at the maximum height.
We know that the bullet has left with a velocity of 300 m/s, and the block has moved from rest to its maximum height, so V' is the final velocity of the block just before it comes to rest momentarily. This final velocity is also equal to v' at its lowest point, as the potential energy is again converted to kinetic energy.
Final kinetic energy = (1/2)(0.01 kg)(300 m/s)^2 + (1/2)(2.0 kg)(4.48 m/s)^2 = 976.43 J
Equating the initial and final energies, we find:
mgh = 976.43 J
h = 976.43 J / (2 kg)(9.81 m/s^2) = 49.83 m
However, this maximum height is measured from the lowest point of the swing. To find the maximum height above the initial position, we subtract the length of the string:
Max height above initial position = h - 1.5 m = 48.33 m
Rounding to the nearest whole centimeter gives us the final answer of: 46 cm.

A Pendulum Consists of a 2