Two equal spheres of mass m are in contact on a smooth horizontal table. A third identical sphere impinges symmetrically on them and is reduced to rest. Find the loss of K.E
16mu2
Let u = velocity of sphere A before impact. As the sphere are identical, the triangle ABC formed by joining their centres is equilateral. The spheres B & C will move in direction AB and AC after impact making an angle of 30∘ with the original lines of motion of ball A.
Let v = speed of the other balls after impact
Momentum Conservation:
mu = mv cos 30∘ + mv cos 30∘
u = v√3 ........(i)
for an oblique collision, we have to take components along normal i.e., along AB for balls A and B
vB−vA = −e(uB−uA) ⇒ v−0 = −e(0−u cos 30∘)
v = eucos 30∘
combining (i) and (ii), we get e = 23.
Loss in KE = 12mu2−2(12mv2) = 12mu2−m(u√3)2 = 16mu2