Two manned satellites approach one another at a relative velocity of v = 0.150 m/s, intending to dock. the first has a mass of 4.00 × 103 kg, and the second a mass of 7.50 × 103 kg. if the two satellites collide elastically rather than dock, what is their final relative velocity?

Refer to the diagram shown below.

The relative velocity before the collision is 0.15 m/s. Therefore consider one satellite to be at rest and the other satellite to have a velocity of 0.15 m/s.
The total momentum before impact is
P₁ = (4000 kg)*(0.15 m/s) = 600 kg-(m/s)

After elastic impact, the rebound velocities are 0.15 m/s and u m/s.
The total momentum after impact is
P₂ = (4000 kg)*(-0.15 m/s) + (7500 kg)*(u m/s) = -600 + 7500u kg-(m/s)

Because momentum is conserved, therefore
-600 + 7500u = 600
u = 1200*7500 = 0.16 m/s

The relative velocity after impact is
0.16 - (-0.15) = 0.31 m/s

Note: In elastic impact, there is no loss of kinetic energy.

0.31 m/s

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