A bullet of mass m is shot at speed v toward a pendulum bob of mass M. The bullet penetrates through the block and emerges on the other side traveling at speed v/2.
1. What is the speed v_block of the block immediately after the bullet emerges from the block (In terms of M, m and v)?
2. To what maximum height h does the block rise (In terms of M, m and v)?
To find the speed of the block immediately after the bullet emerges, we can apply the principle of conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision.
First, let's calculate the momentum of the bullet before the collision. The momentum of an object is given by the product of its mass and velocity.
Momentum of the bullet before the collision = mass of bullet * velocity of bullet
= m * v
Since the bullet emerges from the block with a velocity of v/2, the momentum of the bullet after the collision is given by:
Momentum of the bullet after the collision = mass of bullet * velocity of bullet after the collision
= m * (v/2)
= (m * v) / 2
Now, let's calculate the momentum of the block after the collision. The block initially has a velocity of 0, so its momentum before the collision is 0. After the collision, the block and bullet move together at a velocity v_block. Therefore, the momentum of the block after the collision is given by:
Momentum of the block after the collision = mass of block * velocity of block after the collision
= M * v_block
According to the principle of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision. Therefore, we can equate these two values:
m * v = (m * v) / 2 + M * v_block
Now, we can solve for v_block:
m * v - (m * v) / 2 = M * v_block
(2m * v - m * v) / 2 = M * v_block
m * v / 2 = M * v_block
v_block = (m * v) / (2 * M)
Therefore, the speed of the block immediately after the bullet emerges from the block is (m * v) / (2 * M).
To find the maximum height h to which the block rises, we can use the principle of conservation of mechanical energy, assuming no energy is lost to other factors such as friction or air resistance.
The initial mechanical energy of the system (block and bullet) is given by:
Initial mechanical energy = kinetic energy of the bullet + kinetic energy of the block
= (1/2) * m * v^2 + (1/2) * M * v_block^2
The final mechanical energy of the system is given by:
Final mechanical energy = potential energy of the block at maximum height
= M * g * h
Since energy is conserved, we can equate the two:
(1/2) * m * v^2 + (1/2) * M * v_block^2 = M * g * h
Now, we substitute the value of v_block we found earlier:
(1/2) * m * v^2 + (1/2) * M * ((m * v) / (2 * M))^2 = M * g * h
Simplifying the equation:
(1/2) * m * v^2 + (1/2) * M * (m^2 * v^2) / (4 * M^2) = M * g * h
[(1/2) * m * v^2 * (1 + (m^2) / (4 * M^2))] = M * g * h
[(2m^2 + m^2) * v^2] / (8M) = M * g * h
[3m^2 * v^2] / (8M) = M * g * h
h = [3m^2 * v^2] / (8M^2 * g)
Therefore, the maximum height h to which the block rises is [3m^2 * v^2] / (8M^2 * g).