Speed of person firing gun on ice

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The discussion focuses on calculating the recoil speeds of a person firing a gun on ice, where friction is negligible. When a standard cartridge is fired, a bullet of 17g moves at 250 m/s, while the recoil speed is denoted as vc. For a blank cartridge, a mass of 0.17g is shot at 57 m/s, with the recoil speed represented as vb. To find the ratio vb/vc, the principle of conservation of momentum is applied, using the equations Pi = Pf for both scenarios and then dividing the results. The conversation emphasizes the importance of momentum conservation in determining the recoil speeds without needing to calculate their exact values.
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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!
 
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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.
 

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