You are hired as ballistics expert and need to measure the muzzle speed of the bullet (with a mass, m = 20 g) for the gun. You fire a ballistic gun horizontally into a 2-kg ballistic pendulum (M = 2 kg) hanging at rest on a massless rod. After the bullet hits it and becomes embedded in it, the pendulum swings 0.5 m above its original height (h = 0.5 m).
Find the speed with which the bullet hits the block (muzzle velocity). Round the answer (use no digits after decimal place).
Hint: use both laws of conservation: for energy and momentum.
To find the muzzle velocity of the bullet, we can use the laws of conservation of energy and momentum.
1. Conservation of Energy:
The initial kinetic energy of the bullet is given by (1/2)m*v^2, where m is the mass of the bullet and v is its velocity. The initial potential energy of the pendulum is zero since it is at rest. After the bullet hits and becomes embedded in the pendulum, the system (bullet+pendulum) will reach its maximum height, h.
At this maximum height, all the initial kinetic energy of the bullet has been converted into potential energy of the system:
(1/2)m*v^2 = (M+m)gh
where M is the mass of the pendulum, g is the acceleration due to gravity, and h is the maximum height the pendulum reaches.
2. Conservation of Momentum:
The initial momentum of the bullet is given by mv, and after the bullet becomes embedded in the pendulum, the system (bullet+pendulum) will have a combined mass of (M+m). Since there is no external force acting horizontally on the system, the initial momentum of the bullet is equal to the final momentum of the system:
mv = (M+m)V
where V is the final velocity of the system after the collision.
To solve for the muzzle velocity (v), we need to eliminate V from the above equations. Rearranging the second equation, we get:
V = mv / (M+m)
Substituting this expression for V in the first equation, we have:
(1/2)m*v^2 = (M+m)gh
(1/2)*v^2 = (M+m)gh / m
v^2 = 2(M+m)gh / m
v = sqrt (2(M+m)gh / m)
Substituting the given values:
M = 2 kg
m = 20 g = 0.02 kg
h = 0.5 m
g = 9.8 m/s^2
v = sqrt (2(2+0.02)(9.8)(0.5) / 0.02)
v = sqrt (39.2)
Rounding to the nearest whole number, the muzzle velocity of the bullet is 6 m/s.