A 0.030-kg bullet is fired straight up at a falling wooden block that has a mass of 5.0 kg. The bullet has a speed of 700 m/s when it strikes the block. The block originally was dropped from rest from the top of a building and had been falling for a time t when the collision with the bullet occurs. As a result of the collision, the block (with the bullet in it) reverses direction, rises, and comes to a momentary halt at the top of the building. Find the time t

To find the time 't' at which the collision occurs, we can use the principles of conservation of momentum and energy.

1. Conservation of momentum:
The momentum before the collision is equal to the momentum after the collision.

Momentum before collision = Momentum after collision

The momentum before the collision is the momentum of the bullet, which can be calculated by multiplying its mass by its velocity:

Momentum of bullet before collision = mass of bullet * velocity of bullet

Momentum of bullet before collision = 0.030 kg * 700 m/s

Momentum of bullet before collision = 21 kg·m/s

The momentum after the collision is the momentum of the combined bullet and wooden block system. Since they reverse direction and come to a momentary halt at the top of the building, the momentum after the collision is zero:

Momentum of bullet and block after collision = 0 kg·m/s (since momentum is zero)


So, using the conservation of momentum equation, we can set the momentum before the collision equal to the momentum after the collision:

Momentum before collision = Momentum after collision

21 kg·m/s = 0 kg·m/s


2. Conservation of energy:
The total energy before the collision (kinetic energy of the bullet) is equal to the total energy after the collision (potential energy of the bullet and block at the top of the building).

The kinetic energy of the bullet before the collision can be calculated using the formula:

Kinetic energy of bullet before collision = (1/2) * mass of bullet * (velocity of bullet)^2

Kinetic energy of bullet before collision = (1/2) * 0.030 kg * (700 m/s)^2

Kinetic energy of bullet before collision = 7350 J

At the top of the building, the entire system (bullet and block) comes to a momentary halt, so all the energy is converted into potential energy. The potential energy of the system at the top of the building is given by:

Potential energy of system at top = mass of system * gravitational acceleration * height

The mass of the system is the sum of the mass of the bullet and the mass of the block:

Mass of system = mass of bullet + mass of block

Mass of system = 0.030 kg + 5.0 kg

Mass of system = 5.03 kg

The gravitational acceleration is approximately 9.8 m/s^2, and the height at the top of the building is not given.

Now, we can set the kinetic energy before the collision equal to the potential energy at the top of the building:

Kinetic energy before collision = Potential energy at top

7350 J = 5.03 kg * 9.8 m/s^2 * height

Solving for height gives:

height = (7350 J) / (5.03 kg * 9.8 m/s^2)
= 147.311 m

3. Finding the time 't':
The time 't' at which the collision occurs can be found using the equation of motion for a freely falling object:

height = (1/2) * g * t^2

Rearranging the equation and solving for 't' gives:

t^2 = (2 * height) / g

t = sqrt((2 * 147.311 m) / 9.8 m/s^2)

t ≈ 5.77 s

Therefore, the time 't' at which the collision occurs is approximately 5.77 seconds.