Question

A 9.00-g bullet embeds itself in a 1.50-kg pendulum bob. After the collision, what is the vertical height gained by the pendulum bob?

a) 0.13 m

b) 0.20 m

c) 0.27 m

d) 0.35 m

Answer 1

The vertical height gained by the pendulum bob after the collision is approximately 0.00721 m.

Step-by-step explanation:

To find the vertical height gained by the pendulum bob after the collision, we need to apply the principle of conservation of momentum. The momentum before the collision is equal to the momentum after the collision. The momentum of the bullet is given by its mass multiplied by its velocity, and the momentum of the pendulum bob is given by its mass multiplied by its velocity. Since the bullet embeds itself in the pendulum bob, their velocities become the same after the collision.

First, let's calculate the initial momentum of the bullet:

Initial momentum of the bullet = mass of the bullet x velocity of the bullet

Initial momentum of the bullet = 0.009 kg x 200 m/s = 1.8 kg · m/s

Since the bullet embeds itself in the pendulum bob, their velocities become the same after the collision. Let's assume the common velocity after the collision as 'v'.

Using the principle of conservation of momentum, we can write:

Initial momentum of the bullet = Final momentum of the pendulum bob

1.8 kg · m/s = (mass of the pendulum bob + mass of the bullet) x v

1.8 kg · m/s = (1.5 kg + 0.009 kg) x v

1.8 kg · m/s = 1.509 kg x v

Solving for 'v', we get:

v = 1.8 kg · m/s / 1.509 kg = 1.192 m/s

Now, we can use the principle of conservation of mechanical energy to find the vertical height gained by the pendulum bob.

Initial mechanical energy = Final mechanical energy

The initial mechanical energy is given by the sum of the gravitational potential energy and the kinetic energy of the bullet:

Initial mechanical energy = m · g · h + 0.5 · m · v²

Final mechanical energy is given by the gravitational potential energy of the pendulum bob at its maximum height:

Final mechanical energy = m · g · h'

Since the bullet embeds itself in the pendulum bob, the final velocity of the pendulum bob is zero at its maximum height.

Therefore, we can write:

m · g · h + 0.5 · m · v² = m · g · h'

0.5 · m · v² = m · g · (h' - h)

0.5 · 1.509 kg · (1.192 m/s)² = 1.509 kg x 9.8 m/s² · (h' - h)

0.5 · 1.509 kg · 1.420 m² = 1.509 kg x 9.8 m/s² · (h' - h)

0.1066 = 14.7591 x (h' - h)

Solving for (h' - h), we get:

h' - h = 0.1066 / 14.7591

h' - h ≈ 0.00721 m

Therefore, the vertical height gained by the pendulum bob is approximately 0.00721 m.