5. (a) A small smooth sphere P, of mass m, moving with speed 2u collides directly...

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5. (a) A small smooth sphere P, of mass m, moving with speed 2u collides directly with a small smooth sphere Q, of mass 2m, moving in the same direction with speed u. 2u 2 m 2m The coefficient of restitution between the spheres is 2 Q then collides with a smooth vertical wall which is perpendicular to the direction of motion of the spheres. The coefficient of restitution between Q and the wall is 4 The first collision between the spheres occurred at a distance of 2 m from the wall. Find the distance from the wall at which the next collision between the spheres occurs. (b) A smooth sphere A, of mass 3m, moving with speed u collides obliquely with a smooth sphere B, of mass m, which is at rest. Before the collision, the velocity of A makes an angle a with the line of centres at impact, where 0°<as90°. The velocity of A is deflected through an angle 0 by the collision, so that its velocity after impact makes an angle 0 +a with the line of centres at impact. The coefficient of restitution between the spheres is 3 tan a Show that tan 0 = (i) 2+3tan? a If a = 30°, find the fraction of kinetic energy lost due to the collision. (ii) 5. (a) A small smooth sphere P, of mass m, moving with speed 2u collides directly with a small smooth sphere Q, of mass 2m, moving in the same direction with speed u. 2u 2 m 2m The coefficient of restitution between the spheres is 2 Q then collides with a smooth vertical wall which is perpendicular to the direction of motion of the spheres. The coefficient of restitution between Q and the wall is 4 The first collision between the spheres occurred at a distance of 2 m from the wall. Find the distance from the wall at which the next collision between the spheres occurs. (b) A smooth sphere A, of mass 3m, moving with speed u collides obliquely with a smooth sphere B, of mass m, which is at rest. Before the collision, the velocity of A makes an angle a with the line of centres at impact, where 0°<as90°. The velocity of A is deflected through an angle 0 by the collision, so that its velocity after impact makes an angle 0 +a with the line of centres at impact. The coefficient of restitution between the spheres is 3 tan a Show that tan 0 = (i) 2+3tan? a If a = 30°, find the fraction of kinetic energy lost due to the collision. (ii)