Three objects A, B and C are kept in a straight line on a frictionless horizontal surface. These have masses m, 2m and m, respectively. The object A moves towards B with a speed $9 m/s$ and makes an elastic collision with it. Thereafter, B makes completely inelastic collision with C. All motions occur on the same straight line. Find the final speed (in $m/s)$ of the object C.

Step 1: Identify the initial parameters.

The masses and initial velocities of the three objects are given as follows.

$$m_A = m, \quad m_B = 2m, \quad m_C = m$$ $$u_A = 9\,\text{m/s}, \quad u_B = 0, \quad u_C = 0$$

Step 2: Calculate the velocity of $B$ after the elastic collision with $A$.

Using the standard formula for a one-dimensional elastic collision, the velocity of $B$ after the collision, $v_B'$, is determined.

$$v_B' = \frac{2m_A}{m_A + m_B} u_A$$ $$v_B' = \frac{2m}{m + 2m} (9)$$ $$v_B' = 6\,\text{m/s}$$

Step 3: Analyze the perfectly inelastic collision between $B$ and $C$.

Since $B$ and $C$ undergo a completely inelastic collision, they stick together and move with a common final velocity $v_f$.

$$m_B v_B' + m_C u_C = (m_B + m_C) v_f$$ $$(2m)(6) + 0 = (2m + m) v_f$$ $$12m = 3m v_f$$ $$v_f = 4\,\text{m/s}$$

Step 4: Determine the final speed of object $C$.

The final speed of object $C$ is equal to the common velocity of the combined mass.

$$v_C = 4\,\text{m/s}$$

Answer: $4\,\text{m/s}$