A railway carriage of mass `M_(c)` filled with sand of mass `M_(s)` moves along the rails. The carriage is given an impulse and it starts with a velocity `v_(0)`. At the same time it is observed that the sand starts leaking through a hole at the bottom of the carriage at a constant mass rate `lambda`. Find the distance at which the carriage becomes empty and the velocity attained by the carriage at that time. (Neglect the friction along the rails).

Explanation

To solve the problem, we need to analyze the motion of the railway carriage filled with sand as it loses mass due to the sand leaking out. We will use the principles of conservation of momentum and kinematics.

### Step-by-step Solution:

1. **Initial Conditions**:
- The initial mass of the carriage is \( M_c + M_s \), where \( M_c \) is the mass of the carriage and \( M_s \) is the mass of the sand.
- The initial velocity of the carriage is \( v_0 \).
- The sand leaks out at a constant mass rate \( \lambda \).
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