Further Mechanics A Level Further Maths Worksheets, Questions and Revision

Two smooth spheres A (mass 3 kg) and B (mass 2 kg) move towards each other in a straight line, A at 6 m/s and B at 2 m/s, and collide directly. The coefficient of restitution between them is 0.5. Find the speed and direction of each sphere after the collision, and the magnitude of the impulse exerted by A on B.

1. Take the direction of A's initial velocity as positive, so u_A = 6 m/s and, since B moves towards A (the opposite way), u_B = -2 m/s.
2. Conservation of momentum: 3(6) + 2(-2) = 3v_A + 2v_B, so 14 = 3v_A + 2v_B.
3. Newton's law of restitution: v_B - v_A = -e(u_B - u_A) = -0.5(-2-6) = -0.5(-8) = 4, so v_B - v_A = 4.
4. Substitute v_B = v_A + 4 into the momentum equation: 14 = 3v_A + 2(v_A+4) = 5v_A + 8, giving v_A = 6/5 = 1.2 m/s, and v_B = 5.2 m/s.
5. Check: both velocities are positive (both spheres now move in A's original direction) with v_B > v_A, so B moves away from A and they do not overlap; momentum check 3(1.2)+2(5.2)=3.6+10.4=14 confirms the arithmetic.
6. Impulse exerted by A on B = change in momentum of B = 2(v_B - u_B) = 2(5.2-(-2)) = 2(7.2) = 14.4 N s, in A's original direction of motion.