Speed of person firing gun on ice

[mikefitz]

A person is standing on a sheet of ice so slippery that friction may be ignored. This individual fires a gun parallel to the ground. When a standard cartridge is used , a 17-g bullet is shot forward with a speed of 250 m/s, and the person recoils with a speed of vc. When a blank cartridge is used , a mass of 0.17g is shot forward with a speed of 57 m/s , and the recoil speed is vb. Find the ratio vb/vc.

I'm trying to figure out how to apply the conversation of momentum to this problem. I know that Pi = Pf, but in the case of this problem how would I figure out what 'recoil speed' the person experiences? Please help me begin this problem, thanks!

 

[neutrino]

You don't need to find the speeds, only their ratio is required. Use Pi = Pf in both cases, and then divide one equation by the other.

 

[kyle8921]

I would approach this problem by first considering the conservation of momentum. This principle states that in a closed system, the total momentum before an event is equal to the total momentum after the event.

In this case, the person, gun, and bullet can be considered a closed system. The initial momentum of the system is zero, as the person and gun are at rest. When the bullet is fired, it gains momentum in one direction, causing the person and gun to recoil in the opposite direction. The total momentum of the system after the event is also zero, as the person and gun are now moving in the opposite direction of the bullet.

Using the equation for conservation of momentum, we can set the initial and final momentum of the system equal to each other and solve for the unknown recoil speed (vc or vb).

For the standard cartridge, we have:

(17 g)(250 m/s) = (17 g + m)(vc)

Where m is the mass of the person and gun (which we can assume to be much larger than the bullet).

Solving for vc, we get:

vc = (17 g)(250 m/s) / (17 g + m)

Similarly, for the blank cartridge, we have:

(0.17 g)(57 m/s) = (0.17 g + m)(vb)

Solving for vb, we get:

vb = (0.17 g)(57 m/s) / (0.17 g + m)

To find the ratio vb/vc, we can substitute the expressions for vc and vb into the equation:

vb/vc = [(0.17 g)(57 m/s) / (0.17 g + m)] / [(17 g)(250 m/s) / (17 g + m)]

Simplifying and cancelling out the mass term, we get:

vb/vc = (0.17)(57) / (17)(250) = 0.002.

Therefore, the ratio of the recoil speeds is approximately 0.002. This means that the person experiences a much lower recoil speed when firing the blank cartridge compared to the standard cartridge.

In conclusion, by applying the principle of conservation of momentum, we can determine the ratio of the recoil speeds experienced by a person firing a gun on ice with a standard cartridge and a blank cartridge. This ratio is significantly lower for the blank cartridge, as expected due to its lower bullet mass and velocity.