A rifle with a weight of 40 N fires a 4.0 g bullet with a speed of 300 m/s.
(a) Find the recoil speed of the rifle.
m/s
(b) If a 650 N man holds the rifle firmly against his shoulder, find the recoil speed of the man and rifle.
m/s
To find the recoil speed of the rifle, we can use the principle of conservation of linear momentum. According to this principle, the total momentum before an event is equal to the total momentum after the event, assuming no external forces act on the system.
For part (a), we consider the rifle as the system. The rifle starts at rest, so its initial momentum is zero. The bullet is fired with a certain velocity, let's call it Vb. Considering that the mass of the bullet is 4.0 g (or 0.004 kg), we can calculate the momentum of the bullet as:
Momentum of bullet = mass of bullet * velocity of bullet = 0.004 kg * 300 m/s = 1.2 kg·m/s
According to the law of conservation of momentum, the total initial momentum of the system is equal to the total final momentum. Since the bullet is fired forward, the rifle will experience a recoil in the opposite direction. Let's call the recoil velocity of the rifle Vr. The mass of the rifle is not given, but we can calculate it using Newton's second law, which states that force equals mass times acceleration:
Force = mass * acceleration
The force responsible for the recoil is the reaction force exerted by the bullet. This force can be calculated using Newton's third law, which states that every action has an equal and opposite reaction. The force exerted by the bullet on the rifle is equal in magnitude and opposite in direction to the force exerted by the rifle on the bullet.
Force exerted by the bullet = - Force exerted by the rifle
Given that the force exerted by the bullet is equal to the product of the mass of the bullet and its acceleration (we'll ignore gravity), we have:
mass of bullet * acceleration of bullet = - mass of rifle * acceleration of rifle
Substituting the mass of the bullet and the velocity of the bullet with the recoil velocity of the rifle, we get:
0.004 kg * 300 m/s = - mass of rifle * Vr
Rearranging the equation to solve for Vr, we find:
Vr = (0.004 kg * 300 m/s) / (-mass of rifle)
Since we don't have the mass of the rifle, we cannot determine its exact recoil velocity.
For part (b), we consider the man and the rifle as a combined system. We can apply the same principle of conservation of momentum. The initial momentum of the system is zero, as both the man and the rifle are at rest. When the rifle is fired, it experiences a recoil velocity (Vr) in the opposite direction, and the man also experiences a recoil velocity (Vm) due to the reaction force.
Let's assume the mass of the man is M (in kg), and his recoil velocity is Vm. The mass of the rifle is still unknown, so let's continue using mass of the rifle (in kg) for now.
The total initial momentum of the system is zero. The rifle's momentum is in the negative direction (opposite to the bullet's momentum), while the man's momentum is in the positive direction:
- mass of rifle * Vr + M * 0 = 0
The total final momentum of the system is the sum of the rifle's momentum and the man's momentum:
- mass of rifle * Vr + M * Vm = 0
From these equations, we can solve for Vm:
M * Vm = mass of rifle * Vr
Vm = (mass of rifle * Vr) / M
Again, we need the mass of the rifle in order to determine the exact recoil velocity of the man and rifle.