The lead female character in the movie Diamonds Are Forever is standing at the edge of an offshore oil rig. As she fires a gun, she is driven back over the edge and into the sea. Suppose the mass of a bullet is 0.009 kg, and its velocity is +735 m/s. Her mass (including the gun) is 53 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?
(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 +735 m/s?
To solve this problem, we can apply the law of conservation of momentum. According to this law, the total momentum before an event equals the total momentum after the event, as long as no external forces are acting.
Let's first determine the initial momentum of the female character and the gun.
The initial momentum (before firing the gun) can be calculated using the formula:
Initial momentum = mass × velocity
In this case, the mass (including the gun) is 53 kg, and since the character is at a stationary position, her initial velocity is 0 m/s. Therefore, the initial momentum is 0 kg m/s.
(a) Now let's determine the momentum of the bullet after being fired. The mass of the bullet is 0.009 kg, and its velocity is +735 m/s. Therefore, the momentum of the bullet is:
Bullet momentum = mass × velocity = 0.009 kg × 735 m/s = 6.615 kg m/s
According to the law of conservation of momentum, the total momentum after the event (firing the gun) should be equal to the initial momentum. Therefore, the recoil momentum of the female character and the gun should be -6.615 kg m/s (negative sign indicates opposite direction).
To find the recoil velocity, we can use the formula:
Recoil velocity = recoil momentum / total mass
Here, the total mass (including the gun) is 53 kg. Therefore,
Recoil velocity = -6.615 kg m/s / 53 kg = -0.1244 m/s
So, the recoil velocity of the female character after firing a single shot, assuming no external force keeps her in place, is approximately -0.1244 m/s. The negative sign indicates that she moves in the opposite direction of the bullet.
(b) Now let's consider the scenario where a blank cartridge is fired, which ejects a mass of 5.0 × 10^-4 kg at a velocity of +735 m/s.
Similar to part (a), we can calculate the momentum of the ejected mass:
Ejected mass momentum = mass × velocity = 5.0 × 10^-4 kg × 735 m/s = 0.3675 kg m/s
Again, according to the law of conservation of momentum, the total momentum after the event should be equal to the initial momentum, which is 0 kg m/s.
So, the recoil momentum of the female character and the gun is:
Recoil momentum = -Ejected mass momentum = -0.3675 kg m/s
Using the same formula as before, the recoil velocity is:
Recoil velocity = recoil momentum / total mass = -0.3675 kg m/s / 53 kg = -0.00692 m/s
Therefore, if she shoots a blank cartridge, the recoil velocity of the female character, assuming no external force keeps her in place, would be approximately -0.00692 m/s. Again, the negative sign denotes the opposite direction to the ejected mass.