There different objective of masses `m_(1) , m_(2)` and `m_(2)` are allowed to fall from rest and from the same point O along three difference frictionless path .The speeds of three objects on reaching the ground will be

Options

A. `(1)/(m_(1)):(1)/(m_(2)):(1)/(m_(3))`
B. `1:1:1`
C. `m_(1): 2m_(2) : 3m_(3)`
D. `m_(1): m_(2): m_(3)`

Explanation

To solve the problem of determining the speeds of the three objects \( m_1, m_2, \) and \( m_3 \) when they reach the ground after falling from the same height \( h \) along different frictionless paths, we can use the principle of conservation of energy.

### Step-by-Step Solution:

1. **Identify the Initial and Final States**:
- The objects are released from rest at a height \( h \).
- Initial velocity \( u = 0 \) for all objects.
- Final velocity \( v \) will be determined when they reach the ground.

2. **Apply the Conservation of Mechanical Energy**:
- The potential energy at the height \( h \) will convert into kinetic energy when the objects reach the ground.
- The potential energy (PE) at height \( h \) is given by:
\[ \text{PE} = mgh \]
- The kinetic energy (KE) when the object reaches the ground is given by:
\[ \text{KE} = \frac{1}{2} mv^2 \]

3. **Set Up the Energy Conservation Equation**:
- At the top (height \( h \)), the total energy is purely potential:
\[ mgh = \frac{1}{2} mv^2 \]

4. **Simplify the Equation**:
- Since mass \( m \) appears on both sides of the equation, we can cancel it out (assuming \( m \neq 0 \)):
\[ gh = \frac{1}{2} v^2 \]

5. **Solve for Final Velocity \( v \)**:
- Rearranging the equation gives:
\[ v^2 = 2gh \]
- Taking the square root of both sides:
\[ v = \sqrt{2gh} \]

6. **Conclusion**:
- The final speed \( v \) of each object when they reach the ground is:
\[ v = \sqrt{2gh} \]
- This result shows that the final speed is independent of the mass of the objects and depends only on the height \( h \) and the acceleration due to gravity \( g \).

7. **Ratio of Speeds**:
- Since all three objects \( m_1, m_2, \) and \( m_3 \) have the same final speed:
\[ v_1 : v_2 : v_3 = 1 : 1 : 1 \]

### Final Answer:
The speeds of the three objects when they reach the ground will be in the ratio \( 1 : 1 : 1 \).

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