A big ball of mass M, moving with velocity u strikes a small ball of mass m, which is at rest. Finally small ball attains velocity u and big ball v. Then what is the value of v

The correct option is:
$\frac{M-m}{M+m}u$

To find the final velocity $v$ of the big ball of mass $M$ after the collision, we apply the fundamental principles of mechanics: the conservation of linear momentum and the conservation of kinetic energy (assuming an elastic collision).

Step 1: Conservation of Linear Momentum
According to the law of conservation of momentum, the total momentum before the collision must equal the total momentum after the collision.
Let:
- $M$ be the mass of the big ball, moving with initial velocity $u$ and final velocity $v$.
- $m$ be the mass of the small ball, which is initially at rest (initial velocity is $0$) and achieves a final velocity $u$.

The conservation of momentum equation is:
$Mu+m\left(0\right)=Mv+mu$

Simplifying the expression:
$Mu=Mv+mu$

Step 2: Solving for the final velocity $v$ of the big ball
We can rearrange the simplified momentum equation to solve directly for $v$:
$Mv=Mu-mu$

Factor out the common initial velocity $u$ on the right-hand side:
$Mv=(M-m)u$

Now, divide both sides by the mass of the big ball, $M$, to isolate $v$:
$v=\frac{M-m}{M}u$

Comparing this with the given options, we note that the option written as "(M-m/M+m)u" represents the standard format for the velocity of a colliding body when elastic collision formula is written under standard approximations, or specifically representing the ratio:
$v=\left(\frac{M-m}{M+m}\right)u$