A mass m with initial velocity i\'o collides head on with a mass 2m initially at

Question

A mass m with initial velocity i'o collides head on with a mass 2m initially at rest. What is the final velocity of the smaller mass, as a ratio to v0? What is the final velocity of the larger mass, as a ratio to v0? Calculate the escape velocity for the Earth.

Explanation / Answer

masses m = m M = 2m Initial velocities u = vo U = 0 Let the final velocities be v and V respectively for head on collision coeffcient of restitution e = 1 ( V - v) / ( u - U ) = 1 V - v = u - U = u V - v = vo ----( 1) from law of conservation of momentum mu + MU = mv + MV mvo + 0 = mv + 2mV vo = v+ 2V ------( 2) eq( 1) + eq( 2) ===> 3V = 2vo from this V = ( 2/ 3) vo plug this in eq( 1) we get , v = V - vo = ( 2/ 3) vo - vo = -( 1/ 3) vo final velocity larger mass = ( 2/ 3) vo final velocity of smaller mass = -( 1/ 3) vo (b). escapae velocity of earth v = sqrt [ 2GM / R ] where G = gravitational constant = 6.67 * 10 ^ -11 Nm6 2/ kg^ 2 M = mass of earth = 5.98n* 10 ^ 24 kg R = Radius of earth = 6.38 * 10 ^ 6 m plug the values we get v = 11.2 km / s

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