A bullet of mass m = 0.025 kg is fired with a speed of vi = 94.00 m/s and hits a block of mass M = 2.10 kg supported by two massless strings. The bullet emerges from the right side of the block with a speed of vf = 47.00 m/s. Find the height to which the block rises.
To find the height to which the block rises, we can use the principle of conservation of momentum and the principle of conservation of energy.
1. Conservation of momentum:
The initial momentum before the collision is equal to the final momentum after the collision. Since there are no external horizontal forces acting on the system, the total momentum in the horizontal direction remains constant.
Initial momentum of the bullet = mass of the bullet * initial velocity of the bullet = m * vi
Final momentum of the bullet = mass of the bullet * final velocity of the bullet = m * vf
The total momentum before the collision is the same as the total momentum after the collision:
m * vi = m * vf
2. Conservation of energy:
The total energy before the collision is equal to the total energy after the collision, considering both kinetic energy and potential energy.
Initial kinetic energy of the bullet = (1/2) * mass of the bullet * (initial velocity of the bullet)^2 = (1/2) * m * vi^2
Final kinetic energy of the bullet = (1/2) * mass of the bullet * (final velocity of the bullet)^2 = (1/2) * m * vf^2
Initial potential energy of the block = 0 (as it is at a height of 0 initially)
Final potential energy of the block = mass of the block * acceleration due to gravity * height (since it rises to a height)
The total energy before the collision is the same as the total energy after the collision:
(1/2) * m * vi^2 = (1/2) * m * vf^2 + M * g * h
Now, we can solve the equations simultaneously to find the height to which the block rises.
Substituting the value of vf from equation (1) into equation (2):
(1/2) * m * vi^2 = (1/2) * m * (vi^2/2^2) + M * g * h
(1/2) * m * vi^2 = (1/2) * m * (vi^2/4) + M * g * h
Simplifying the equation:
vi^2 = (vi^2/4) + 4 * M * g * h
Multiply both sides by 4:
4 * vi^2 = vi^2 + 16 * M * g * h
3 * vi^2 = 16 * M * g * h
h = (3 * vi^2) / (16 * M * g)
Now, substitute the known values to calculate the height:
m = 0.025 kg, vi = 94.00 m/s, M = 2.10 kg, g = 9.8 m/s^2
h = (3 * (94.00)^2) / (16 * 2.10 * 9.8) = 7.37 m
Therefore, the height to which the block rises is approximately 7.37 meters.