A bullet of mass 6.45 g is fired from a 66.0 cm-long rifle barrel at 237 m/s. The mass of the
rifle is 4.50 kg. Determine the acceleration of the bullet as well as the recoil acceleration of
the rifle.
V^2 = Vo^2 + 2a*d.
V = 137 m/s, Vo = 0, d = 0.66m, a = ?.
To determine the acceleration of the bullet and the recoil acceleration of the rifle, we can use the principle of conservation of momentum. According to this principle, the total momentum before firing the bullet is equal to the total momentum after firing the bullet.
The total momentum before firing the bullet can be calculated by considering the momentum of the bullet and the momentum of the rifle. The momentum of an object is equal to its mass multiplied by its velocity.
Given:
Mass of the bullet (m1) = 6.45 g = 0.00645 kg
Velocity of the bullet (v1) = 237 m/s
Mass of the rifle (m2) = 4.50 kg
Velocity of the rifle (v2) = 0 (since the rifle is initially at rest)
Step 1: Calculate the momentum of the bullet (p1).
p1 = m1 * v1
= 0.00645 kg * 237 m/s
= 1.52865 kg·m/s
Step 2: Calculate the momentum of the rifle (p2).
p2 = m2 * v2
= 4.50 kg * 0 m/s
= 0 kg·m/s
Step 3: Calculate the total momentum before firing (p_initial).
p_initial = p1 + p2
= 1.52865 kg·m/s + 0 kg·m/s
= 1.52865 kg·m/s
Step 4: Calculate the total momentum after firing (p_final).
Since the bullet is fired, its momentum after firing is 0, and the rifle recoils with a velocity (v_final).
Therefore, p_final = 0.
Step 5: Apply the principle of conservation of momentum.
According to the principle of conservation of momentum,
p_initial = p_final
1.52865 kg·m/s = 0
Since p_final = 0, it implies that the total momentum before firing is completely canceled out by the total momentum after firing.
Step 6: Calculate the recoil velocity of the rifle (v_final).
v_final = p_initial / m2
= 1.52865 kg·m/s / 4.50 kg
≈ 0.3397 m/s
Step 7: Calculate the acceleration of the bullet (a_bullet).
The acceleration of the bullet can be found using Newton's second law, F = ma, where F is the force and m is the mass of the bullet.
Since we know that the force acting on the bullet is the force exerted by the expanding gases, which is equal to the mass of the bullet multiplied by its acceleration:
F = m_bullet * a_bullet
Since the force exerted on the rifle is equal in magnitude but opposite in direction:
F = m_rifle * a_rifle
Equating these forces, we can solve for the acceleration of the bullet.
m_bullet * a_bullet = m_rifle * a_rifle
a_bullet = (m_rifle * a_rifle) / m_bullet
Substituting the given values:
a_bullet = (4.50 kg) * (0.3397 m/s) / (0.00645 kg)
≈ 2357.946 m/s²
Therefore, the acceleration of the bullet is approximately 2357.946 m/s², and the recoil acceleration of the rifle is approximately 0.3397 m/s².
To determine the acceleration of the bullet and the recoil acceleration of the rifle, we can apply the principle of conservation of momentum. According to this principle, the total momentum before the bullet is fired is equal to the total momentum after the bullet is fired.
The momentum of an object is given by the product of its mass and velocity:
Momentum = mass × velocity
Let's calculate the initial momentum of the bullet and the rifle:
Initial momentum of bullet = Mass of the bullet × Velocity of the bullet
= 0.00645 kg × 237 m/s
Initial momentum of rifle = Mass of the rifle × Velocity of the rifle
= 4.5 kg × 0 m/s (the rifle is initially at rest)
The total initial momentum is the sum of the individual momenta:
Total initial momentum = Initial momentum of bullet + Initial momentum of rifle
Now, since the principle of conservation of momentum states that the total momentum before and after the event must be the same, we can equate the total initial momentum to the total final momentum. Since the bullet is no longer in the barrel after being fired, the final momentum of the bullet is zero.
Total initial momentum = Total final momentum
(Initial momentum of bullet + Initial momentum of rifle) = Final momentum of rifle
Now, let's solve for the final momentum of the rifle:
Final momentum of rifle = (Initial momentum of bullet + Initial momentum of rifle)
The acceleration of an object is given by the change in its velocity over time. We can use Newton's second law of motion, F = ma, to determine the acceleration. Rearranging the equation, we have a = F/m.
Considering only the forces acting on the bullet and the rifle during this event, we can assume that the forces of interaction between the two objects (bullet and rifle) are internal forces and cancel each other out. Therefore, the only external force acting on the system is the force exerted by the bullet as it is fired.
The force exerted by the bullet can be calculated using the equation F = ΔP/t, where ΔP is the change in momentum and t is the time taken for this change. Since there is no time given in the problem, we can assume that the time taken is very small, so the force is very large and can be considered instantaneous. Therefore, we can consider the force exerted by the bullet as the only external force acting on the system.
Now, let's substitute the values to calculate the acceleration of the rifle:
Force exerted by the bullet = (Initial momentum of bullet + Initial momentum of rifle) / t
Recoil acceleration of the rifle = (Force exerted by the bullet) / Mass of the rifle
Now, you can substitute the given values into the equations to calculate the acceleration of the bullet and the recoil acceleration of the rifle.