In the \"slingshot effect,\" the transfer of energy in an elastic collision is u

Question

In the "slingshot effect," the transfer of energy in an elastic collision is used to boost the energy of a space probe so that it can escape from the solar system. All speeds are relative to an inertial frame in which the center of the Sun remains at rest. The figure below shows a space probe moving at speed u1 = 10.2 km/s toward Saturn, which is moving at speed u2 = 9.9 km/s toward the probe. Because of the gravitational attraction between Saturn and the probe, the probe swings around Saturn and heads back in the opposite direction with speed vf.

(a) Assuming this collision to be a one-dimensional elastic collision with the mass of Saturn much much greater than that of the probe, find vf.
km/s

(b) By what factor is the kinetic energy of the probe increased?


Where does this energy come from?

Explanation / Answer

The speed of the probe toward the saturn is (u1 ) = 10 .2 km /s The speed of the saturn toward the probe is (u2 ) = 9.9  km /s Since, the mass of the saturn is very very greater than that of the prope, the center of mass of the system also moves with the velocity of the saturn . ( v =9.9  km /s ) Let us consider, After collision , the saturn and the probe moving in the positive direction of x axis. The speed of approach relative to the center of mass is                                   u   =  - u1 - u2                                      = - 20.1 km /s In elastic collision, The relative speed of approach , and relative speed of recession are always in opposite direction, Hence, After collision , relative speed of recession is        u ( recession ) = - u                                 =   20.1 km /s Hence, the relative speed of recession and relative speed of center of mass are related as                    vf       =     20.1 km /s    +    9.9  km /s                               =   30 km /s ------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------- (b) The ratio of final and inital kinetic energies of the system is               Kf / Ki    =    ( ( 1/2) M ( vf   )2 )   / ( 1/2) M  (  u1 ) )2                                = (   30 km /s ) 2   / ( 10.2 km /s )2                                    = 8.65 Final kinetic energy is greater than the inital energy . This energy comes from small slowing down of the saturn.                                = (   30 km /s ) 2   / ( 10.2 km /s )2                                    = 8.65 Final kinetic energy is greater than the inital energy . This energy comes from small slowing down of the saturn.

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