A physics teacher shoots a .30 caliber rifle at a 0.47 kg block of wood. The rifle and wood are mounted on separate carts that sit atop an air track (like a linear air hockey table--ie. frictionless). The 5.7 kg rifle fires a 27 gram bullet at 227 m/s in the positive direction.
What would be the velocity of the rifle after the bullet is fired?
And, what would be the velocity of the block of wood with the bullet lodged inside?
V = 1.20 m/s
Vfinal = 12.33 m/s
The total momentum of gun and bullet after firing remains zero. That means the momentum of the bullet and the gun are equal and opposite in sign.
Let V be the recoil velocity of the gun
5.7*V = 0.027*227
Solve for V.
Momemtum of bullet and block are the same before and after impact.
227*0.027 = (0.47 +0.027)*Vfinal
Solve for Vfinal/
To find the velocity of the rifle after the bullet is fired, we need to apply the principle of conservation of momentum. According to this principle, the total momentum before the bullet is fired should be equal to the total momentum after the bullet is fired.
Given:
- Mass of the rifle (m1) = 5.7 kg
- Mass of the bullet (m2) = 0.027 kg
- Initial velocity of the bullet (v2) = 227 m/s
Let's denote the velocity of the rifle (v1) and the velocity of the bullet and block (vf).
The initial momentum of the system (before the bullet is fired) is:
P_initial = m1 * v1 + m2 * 0
= m1 * v1 + 0
The final momentum of the system (after the bullet is fired) is:
P_final = (m1 + m2) * vf
According to the conservation of momentum, P_initial = P_final.
Therefore, m1 * v1 = (m1 + m2) * vf
Now we can solve for the velocity of the rifle (v1):
m1 * v1 = (m1 + m2) * vf
5.7 kg * v1 = (5.7 kg + 0.027 kg) * vf
v1 = (5.7 kg + 0.027 kg) * vf / 5.7 kg
v1 = 0.9998 * vf
Now we can find the velocity of the block of wood (vf) with the bullet lodged inside it. According to the conservation of momentum, the momentum of the bullet and block of wood together after the bullet is fired should be equal to the initial momentum of the system.
Therefore, (m1 + m2) * vf = m1 * v1
(5.7 kg + 0.027 kg) * vf = 5.7 kg * 0.9998 * vf
Simplifying,
(5.727 kg) * vf = 5.6976 kg * vf
The velocity of the block of wood with the bullet lodged inside, vf, is the same as the velocity of the rifle, v1, which we previously found.
Therefore, the velocity of the rifle after the bullet is fired is approximately 0.9998 times the velocity of the bullet, or 226.9176 m/s.
Also, the velocity of the block of wood with the bullet lodged inside would be approximately the same, so it would also be 226.9176 m/s.
To find the velocity of the rifle after the bullet is fired, we can use the conservation of momentum. According to this principle, the total momentum before an event is equal to the total momentum after the event, assuming there are no external forces acting on the system.
Let me explain how to calculate it step by step:
Step 1: Calculate the initial momentum of the system.
The initial momentum of the system is the sum of the momentum of the rifle and the momentum of the block of wood before the bullet is fired.
Momentum (p) is defined as the mass (m) of an object multiplied by its velocity (v).
The momentum of the rifle before the bullet is fired can be calculated as:
Rifle momentum before = mass of the rifle (5.7 kg) multiplied by its initial velocity (0 m/s) since the rifle is initially at rest.
Step 2: Calculate the momentum of the bullet.
The momentum of the bullet can be calculated as:
Bullet momentum before = mass of the bullet (27 g = 0.027 kg) multiplied by its initial velocity (227 m/s) in the positive direction.
Step 3: Calculate the final momentum of the system.
The final momentum of the system is the sum of the momentum of the rifle and the momentum of the bullet after the bullet is fired. Since the bullet and the rifle are initially at rest, the velocity of the bullet embedded in the block of wood would be the same as the velocity of the block. Let's call this final velocity "v_f".
The final momentum of the system can be calculated as:
Final momentum = (mass of the rifle + mass of the bullet) multiplied by the final velocity (v_f).
Step 4: Apply the conservation of momentum.
According to the conservation of momentum, the initial momentum of the system is equal to the final momentum of the system.
Therefore, we can set up the equation:
Rifle momentum before + Bullet momentum before = Final momentum.
By substituting the values we have calculated into the equation, we can solve for the final velocity of the rifle as well as the block of wood with the bullet lodged inside.