A 0.014 kg bullet is fired straight up at a falling wooden block that has a mass of 4.0 kg. The bullet has a speed of 760 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 occured. 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, we can start by using the principle of conservation of momentum. The momentum before the collision is equal to the momentum after the collision.
Before the collision:
Momentum of the bullet = mass of the bullet × velocity of the bullet
Momentum of the block = mass of the block × velocity of the block
After the collision:
Since the bullet and the block reverse direction and come to a momentary halt at the top of the building, the velocities of both the bullet and the block become zero.
Using the principle of conservation of momentum:
Momentum before the collision = momentum after the collision
(mass of the bullet × velocity of the bullet) + (mass of the block × velocity of the block) = 0
Substituting the given values:
(0.014 kg × 760 m/s) + (4.0 kg × velocity of the block) = 0
Now, solve this equation for the velocity of the block, which will be the negative of the velocity of the bullet. Once you have the velocity of the block, you can use it to find the time t by using the equation of motion for uniformly accelerated motion:
velocity of the block = initial velocity + (acceleration × time)
In this case, the initial velocity is zero (since the block was dropped from rest), and the acceleration is due to the force applied by the bullet.
So, the equation becomes:
velocity of the block = 0 + (acceleration × time)
Solving for time t will give you the answer.