An astronaut with mass M in space has a spring-loaded pellet gun. It fires pelle

Question

An astronaut with mass M in space has a spring-loaded pellet gun. It fires pellets of mass m, giving them a muzzle velocity of vp. He wants to use it to get back to his ship, which he is some distance away from. He only has 2 pellets. He loads the gun points it directly away from the ship, fires a pellet, then loads the next and fires it again. Initially, his velocity relative to the ship is 0 m/s. Find an expression for his speed in terms of m, M, and vp relative to the ship after firing both pellets.

Explanation / Answer

for 1st time.

initial momentum is zero.

and after firing bullet is moving with Vp in opposite direction.

0 = m * -Vp + Mv

v = m Vp / M

for 2nd time.

initial speed of astronaut = v = m Vp/ M

initial speed of bullet = 0

final speed of bullet (wrt to ship) = vp -v = (vp - mvP / M) = Vp ( M - m) / M

using momentum conservation,

M x m VP / M   +mx 0   =   - m ( Vp ( M - m) / M ) + Mv

(m M Vp + m M Vp - m^2 VP / M ) = Mv

v = 2 m MVp/M^2 - m^2 Vp/M^2   = mVp ( 2M - m ) /M^2

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