The ballistic pendulum is a device used to measure the speed of a fast-moving projectile such as a bullet. The bullet is fired into a large block of wood suspended from some light wires. The bullet is stopped by the block, and the entire system swings up to a height h. It is possible to obtain the initial speed of the bullet by measuring h and the two masses. Assuming that the mass of the bullet is 5.00g, the mass of the pendulum is 1.00kg, and h id 5.oocm. Find the intial speed of the bullet. (v1=199 m/s)
To find the initial speed of the bullet, we can use the principle of conservation of momentum and the principle of conservation of mechanical energy.
1. Conservation of Momentum:
The initial momentum of the bullet-block system is zero since the block is initially at rest. After the collision, the block and the bullet move together with a common velocity. So, we can write:
(mass of bullet) * (velocity of bullet) = (mass of bullet + mass of block) * (common velocity)
Substituting the given values:
(0.005 kg) * v1 = (0.005 kg + 1.00 kg) * v2
2. Conservation of Mechanical Energy:
The total mechanical energy of the system is conserved. The initial energy is in the form of gravitational potential energy, given by mgh, and the final energy is in the form of kinetic energy, given by (1/2) * (mass of bullet + mass of block) * (common velocity)^2. So, we can write:
Initial energy = Final energy
(mass of bullet + mass of block) * g * h = (1/2) * (mass of bullet + mass of block) * v2^2
Substituting the given values:
(0.005 kg + 1.00 kg) * 9.8 m/s^2 * 0.050 m = (1/2) * (0.005 kg + 1.00 kg) * v2^2
Simplifying the equation, we get:
0.049 kg * 9.8 m/s^2 * 0.050 m = 0.5025 kg * v2^2
0.02401 kg m^2/s^2 = 0.5025 kg * v2^2
v2^2 = (0.02401 kg m^2/s^2) / 0.5025 kg
v2^2 = 0.0478 m^2/s^2
v2 = √(0.0478 m^2/s^2)
Now, we can substitute the value of v2 back into the momentum equation to solve for v1:
(0.005 kg) * v1 = (0.005 kg + 1.00 kg) * √(0.0478 m^2/s^2)
v1 = ((0.005 kg + 1.00 kg) * √(0.0478 m^2/s^2)) / 0.005 kg
Simplifying further:
v1 = (1.005 kg * √(0.0478 m^2/s^2)) / 0.005 kg
v1 = 1.005 * √(0.0478 m^2/s^2) kg / kg
v1 = 1.005 * √(0.0478 m^2/s^2)
Calculating the value:
v1 = 1.005 * 0.218 m/s
v1 ≈ 0.219 m/s
Therefore, the initial speed of the bullet is approximately 0.219 m/s.