Final answer:
The recoil velocity of the 1.00-kg plunger is -12 m/s, and the average force exerted by the gun to stop the plunger over 20.0 cm is 360 N. Compared to the 1200 N exerted on the gun when the bullet is accelerated over 10.0 ms, this demonstrates the effectiveness of recoil reduction mechanisms.
Step-by-step explanation:
Recoil Velocity and Force Calculations
The questions posed involve concepts of momentum conservation and work-energy principles in physics. When dealing with recoil velocity, one takes into account that the total momentum before and after the bullet is fired must be equal, given that momentum is conserved in a closed system. As such, for part (a) we calculate the recoil velocity (v_r) as follows:
Momentum before firing = Momentum after firing
0 = m_bullet * v_bullet + m_plunger * v_r
So, v_r = -(m_bullet * v_bullet) / m_plunger
v_r = -((0.0200 kg * 600 m/s) / 1.00 kg)
v_r = -12 m/s
The negative sign indicates that the plunger moves in the opposite direction to the bullet. For part (b), the work done to stop the plunger can be equated to the change in kinetic energy. The average force (F_avg) can thus be calculated over the stopping distance (d):
F_avg = Work / d = Change in kinetic energy / d
F_avg = (0.5 * m_plunger * v_r^2) / d
F_avg = (0.5 * 1.00 kg * (12 m/s)^2) / 0.20 m
F_avg = 360 N
For part (c), the force applied to the gun if the bullet is accelerated in 10.0 ms can be found by first calculating the acceleration (a) using the kinematic equation:
v = u + at
600 m/s = 0 + a * 0.010 s
a = 60,000 m/s^2
Then, using Newton's second law, the force (F) is:
F = m_bullet * a
F = 0.0200 kg * 60,000 m/s^2
F = 1200 N
Comparing the forces, the average force exerted by the damping mechanism (360 N) is significantly less than the force exerted if the bullet were to be accelerated in 10.0 ms (1200 N), showcasing the importance of such recoil mitigation systems in firearms.