The lead female character in the movie Diamonds Are Forever is standing at the edge of an offshore oil rig. As she fires the gun, she is driven back over the edge and into the sea. Suppose the mass of a bullet is 0.005 kg, and its velocity is +726 m/s. Her mass (including the gun) is 48 kg.
(a) What recoil velocity does she acquire in response to a single shot from a stationary position, assuming that no external force keeps her in place?
m/s
(b) Under the same assumption, what would be her recoil velocity if, instead, she shoots a blank cartridge that ejects a mass of 5.0 10-4 kg at a velocity of +726 m/s?
m/
To find the recoil velocity of the lead female character in both scenarios, we can make use of the law of conservation of momentum, which states that the total momentum before an event is equal to the total momentum after the event, provided that no external forces act on the system.
(a) In the first scenario, the woman is firing a bullet with a mass of 0.005 kg and a velocity of +726 m/s. Her own mass, including the gun, is 48 kg. Let's denote the recoil velocity of the woman as V.
According to the law of conservation of momentum, the initial momentum of the system (before the shot) is zero since the woman is stationary. Therefore, the final momentum of the system (after the shot) must also be zero.
The initial momentum of the bullet is calculated by multiplying its mass and velocity:
Momentum of bullet = Mass of bullet × Velocity of bullet = 0.005 kg × 726 m/s
The final momentum of the system can be calculated by taking into account the bullet's momentum (as it is fired backward) and the woman's momentum (as she is driven back into the sea):
Final momentum = (Mass of bullet × Recoil velocity) + (Mass of woman × Recoil velocity)
Setting the initial momentum equal to the final momentum to satisfy the conservation of momentum, we have:
0 = (0.005 kg × 726 m/s) + (48 kg × Recoil velocity)
Now, we can solve for the recoil velocity:
Recoil velocity = - (0.005 kg × 726 m/s) / (48 kg)
Recoil velocity ≈ -0.076 m/s
Therefore, the woman acquires a recoil velocity of approximately -0.076 m/s (negative indicates the backward direction) in response to a single shot from a stationary position.
(b) In the second scenario, instead of firing a bullet, the woman shoots a blank cartridge that ejects a mass of 5.0 × 10^-4 kg at a velocity of +726 m/s. Let's denote the recoil velocity in this scenario as V'.
Using the same approach as in the previous scenario, we set the initial momentum equal to the final momentum and solve for V':
0 = (5.0 × 10^-4 kg × 726 m/s) + (48 kg × V')
Simplifying and solving for V':
V' = - ((5.0 × 10^-4 kg × 726 m/s) / (48 kg))
V' ≈ -0.0095 m/s
Therefore, in this scenario, the woman acquires a recoil velocity of approximately -0.0095 m/s (negative indicating the backward direction) when shooting a blank cartridge.
Note: In both scenarios, the recoil velocities are relatively small, indicating that the woman's movement would be negligible compared to the forces involved.