Two blocks of masses `m_(1)` and `m_(2)` are connected by a massless pulley A, slides along th esmooth sides of a rectangular wedge of mass m, which rests on a smooth horizontal plane. Find the distance covered by the wedge on the horizontal plane till the mass `m_(1)` is lowered by the vertical distance h.

Explanation

To solve the problem, we will analyze the motion of the two blocks and the wedge. The key is to understand the relationship between the vertical movement of mass \( m_1 \) and the horizontal movement of the wedge.

### Step-by-Step Solution:

1. **Understanding the System**:
- We have two masses \( m_1 \) and \( m_2 \) connected by a pulley. The mass \( m_1 \) is hanging vertically, while \( m_2 \) is on the wedge which is on a smooth horizontal surface. The wedge has a mass \( m \).
- The angle of the wedge with the horizontal is denoted as \( \alpha \).

2. **Vertical Movement of \( m_1 \)**:
- When mass \( m_1 \) is lowered by a vertical distance \( h \), it will cause the wedge to move horizontally.

3. **Horizontal Movement of the Wedge**:
- As \( m_1 \) moves down by \( h \), the length of the string on the side of the wedge also changes. The horizontal component of this movement can be expressed in terms of the angle \( \alpha \).
- The horizontal distance \( x \) moved by the wedge can be related to \( h \) by the geometry of the situation:
\[ x = h \cdot \tan(\alpha) \]

4. **Using the Geometry of the Wedge**:
- The distance \( x \) that the wedge moves horizontally can be derived from the fact that the vertical drop of \( m_1 \) translates into a horizontal movement of the wedge.
- Since the wedge moves horizontally while \( m_1 \) moves vertically, we can use the relationship:
\[ \text{Horizontal distance moved by wedge} = h \cdot \tan(\alpha) \]

5. **Final Expression**:
- Therefore, the distance \( d \) covered by the wedge on the horizontal plane until \( m_1 \) is lowered by the vertical distance \( h \) is given by:
\[ d = h \cdot \tan(\alpha) \]

### Final Answer:
The distance covered by the wedge on the horizontal plane till the mass \( m_1 \) is lowered by the vertical distance \( h \) is \( d = h \cdot \tan(\alpha) \).