A smooth sphere of mass M moving with velocity u directly collides elastically with another sphere of mass m at rest. After collision their final velocities are V and v respectively. The value of v is

To find the final velocity $v$ of the second sphere of mass $m$ after an elastic collision, we can use the principles of conservation of linear momentum and the coefficient of restitution.

Step 1: Conservation of Linear Momentum
The total momentum before the collision must equal the total momentum after the collision. Since the sphere of mass $m$ is initially at rest:
$Mu+m\left(0\right)=MV+mv$
Simplifying this expression gives:
$Mu=MV+mv$
Rearranging the terms, we get:
$M(u-V)=mv$ (Equation 1)

Step 2: Elastic Collision Relationship (Coefficient of Restitution)
For a perfectly elastic collision, the coefficient of restitution is 1, which means the relative velocity of separation is equal to the relative velocity of approach:
$v-V=u-0$
This simplifies to:
$V=v-u$ (Equation 2)

Step 3: Solving for $v$
Substitute Equation 2 into Equation 1:
$M(u-(v-u\left)\right)=mv$
$M(2u-v)=mv$
$2Mu-Mv=mv$
Rearranging to group the terms with $v$:
$2Mu=(M+m)v$
$v=\frac{2Mu}{M+m}$

To match the given options, we divide both the numerator and the denominator of the right-hand side by $M$:
$v=\frac{2u}{1+\frac{m}{M}}$

Therefore, the value of $v$ is $2u/(1+m/M)$.