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 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 occurred. 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 first need to use the principle of conservation of momentum.
The momentum before the collision is equal to the momentum after the collision. Initially, the momentum of the bullet is given by the product of its mass (0.014 kg) and velocity (700 m/s). The momentum of the wooden block (with the bullet inside it) after the collision is zero because it comes to a stop at the top of the building.
Therefore, we can set up the equation:
momentum of bullet before collision = momentum of block after collision
(mass of bullet) * (velocity of bullet) = (mass of block) * (velocity of block)
(0.014 kg) * (700 m/s) = (4.0 kg) * (velocity of block)
Now, to find the velocity of the block, we need to use the equations of motion. Since the block rises, it is moving against gravity, so we will use the equation for vertical motion with constant deceleration:
final velocity^2 = initial velocity^2 + 2 * acceleration * (displacement)
Here, the final velocity is 0 (as the block comes to a halt), and the initial velocity is the velocity of the block at the moment of collision, which is unknown. The acceleration is due to gravity and is approximately -9.8 m/s^2 (negative because it acts against the motion), and the displacement is the height from which the block was dropped.
Since the block comes to a halt at the top of the building, its displacement is equal to the height from which it was dropped.
Using these equations, we can solve for the velocity of the block. Once we have the velocity, we can determine the time t using the equation:
final velocity = initial velocity + acceleration * time
Since the final velocity is 0 when the block comes to a stop, we can solve for time.
By plugging in the numbers and solving the equations, we can find the time t.