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 0.015 kg, and its velocity is +726 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 +726 m/s?
(a) Well, this situation sounds fishy to me, but let's crunch some numbers! According to Newton's third law of motion, for every action, there is an equal and opposite reaction. So, when she fires the gun, the bullet and she both experience a backward force.
To calculate her recoil velocity, we can use the principle of conservation of momentum. The momentum before firing is zero since she is stationary. Therefore, the momentum after firing must also be zero.
Let's denote her recoil velocity as V.
The total initial momentum is given by the equation:
(53 kg) * 0 + (0.015 kg) * (726 m/s) = 0
Now, solving for V:
(53 kg + 0.015 kg) * V = (0.015 kg) * (726 m/s)
V = (0.015 kg) * (726 m/s) / (53 kg + 0.015 kg)
Calculating this value, we find that her recoil velocity is approximately equal to 0.195 m/s.
So, after firing a single shot, she'll be moving backward at about 0.195 m/s. Hold on tight!
(b) Now, if she shoots a blank cartridge, which ejects a mass of 5.0 * 10^(-4) kg at a velocity of +726 m/s, we can use the same principle of conservation of momentum.
The total initial momentum is again zero, since she starts from a stationary position. Therefore, the momentum after firing the blank cartridge must also be zero.
Let's denote her new recoil velocity as V'.
The total final momentum is given by the equation:
(53 kg) * 0 + (5.0 * 10^(-4) kg) * (726 m/s) = 0
Now, solving for V':
(53 kg + 5.0 * 10^(-4) kg) * V' = (5.0 * 10^(-4) kg) * (726 m/s)
V' = (5.0 * 10^(-4) kg) * (726 m/s) / (53 kg + 5.0 * 10^(-4) kg)
After some calculations, we find that her recoil velocity with the blank cartridge is approximately equal to 0.014 m/s. So, it seems like shooting blanks won't have much of an impact on her (pun intended)!
Remember, safety first!
(a) To find the recoil velocity of the female character, we can use the principle of conservation of momentum. The total initial momentum is equal to the total final momentum.
Given:
Mass of the bullet (m1) = 0.015 kg
Velocity of the bullet (v1) = +726 m/s
Mass of the female character and the gun (m2) = 53 kg
Recoil velocity of the female character (v2)
Using the conservation of momentum equation:
(m1 * v1) + (m2 * 0) = (m1 * 0) + (m2 * v2)
Since the initial velocity of the bullet is positive and the final velocity of the bullet and female character are in opposite directions, we can write:
(m1 * v1) = (m2 * -v2)
Solving for v2:
v2 = -((m1 * v1) / m2)
Plugging in the given values:
v2 = -((0.015 kg * 726 m/s) / 53 kg)
v2 ≈ -0.205 m/s
Therefore, the female character acquires a recoil velocity of approximately -0.205 m/s when she fires a single shot.
(b) Similarly, we can apply the same principle of conservation of momentum to calculate the recoil velocity when the character shoots a blank cartridge.
Given:
Mass of the ejected mass (m1) = 5.0 * 10^-4 kg
Velocity of the ejected mass (v1) = +726 m/s
Mass of the female character and the gun (m2) = 53 kg
Recoil velocity of the female character (v2)
Using the conservation of momentum equation:
(m1 * v1) + (m2 * 0) = (m1 * 0) + (m2 * v2)
Again, since the initial velocity of the ejected mass is positive, and the final velocities are in opposite directions, we can write:
(m1 * v1) = (m2 * -v2)
Solving for v2:
v2 = -((m1 * v1) / m2)
Plugging in the given values:
v2 = -((5.0 * 10^-4 kg * 726 m/s) / 53 kg)
v2 ≈ -0.00684 m/s
Therefore, the female character acquires a recoil velocity of approximately -0.00684 m/s when she shoots a blank cartridge.
(a) To find the recoil velocity the lead female character acquires in response to a single shot, we can apply the principle of conservation of momentum. According to this principle, the total momentum before and after the shot must be equal.
The momentum of an object is given by the product of its mass and velocity: momentum = mass × velocity.
The initial momentum before the shot is zero since the lead female character is stationary. Therefore, the momentum after the shot must also be zero for momentum to be conserved.
Let V be the recoil velocity of the lead female character. The momentum after the shot can be expressed as the product of the combined mass of the lead female character and the gun, and the recoil velocity: Momentum = (mass_character + mass_gun) × V.
The momentum contributed by the fired bullet is given by the product of its mass and velocity: Momentum_bullet = mass_bullet × velocity_bullet.
Since the total momentum after the shot must be zero, we can write the equation: (mass_character + mass_gun) × V + mass_bullet × velocity_bullet = 0.
Plugging in the given values: mass_character = 53 kg, mass_bullet = 0.015 kg, velocity_bullet = +726 m/s, we can solve for V.
(53 kg + mass_gun) × V + (0.015 kg) × (726 m/s) = 0.
Simplifying the equation: V = - (0.015 kg × 726 m/s) / (53 kg + mass_gun).
Please note that we don't have enough information about the mass of the gun to calculate the exact value of V. We would need that information to determine the recoil velocity in this scenario.
(b) Similar to part (a), we will use the principle of conservation of momentum to find the recoil velocity when shooting a blank cartridge.
The initial momentum before the shot is still zero since the lead female character is stationary. The momentum after the shot, considering both the character and the gun, must also be zero for momentum to be conserved.
Let V be the recoil velocity of the lead female character. The momentum after the shot can be expressed as the product of the combined mass of the lead female character and the gun, and the recoil velocity: Momentum = (mass_character + mass_gun) × V.
The momentum contributed by the ejected mass (from the blank cartridge) is given by the product of its mass and velocity: Momentum_ejected_mass = mass_ejected_mass × velocity_bullet.
Since the total momentum after the shot must be zero, we can write the equation: (mass_character + mass_gun) × V + mass_ejected_mass × velocity_ejected_mass = 0.
Plugging in the given values: mass_character = 53 kg, mass_ejected_mass = 5.0 × 10^-4 kg, velocity_ejected_mass = +726 m/s, we can solve for V.
(53 kg + mass_gun) × V + (5.0 × 10^-4 kg) × (726 m/s) = 0.
Simplifying the equation: V = - (5.0 × 10^-4 kg × 726 m/s) / (53 kg + mass_gun).
Similar to part (a), we still need information about the mass of the gun to calculate the exact value of V.