A and B participate in a race with acceleration `a_1 and a_2` , respectively . A reaches t times earlier than B at finish line and their velocities at finish line are `v_1 and v_2` , respectively. If difference between their velocities is v , then find the value of v

Options

A. `(a_1+a_2)/2t`
B. `sqrt(a_1a_2)t`
C. `(a_1a_2)/(a_1+a_2)t`
D. `(2a_1a_2)/(a_1+a_2)t`

Answer

To solve the problem step by step, we will use the equations of motion and the relationship between time, acceleration, and velocity.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- A and B are racing with accelerations \( a_1 \) and \( a_2 \) respectively.
- A finishes the race \( t \) seconds earlier than B.
- The final velocities at the finish line are \( v_1 \) for A and \( v_2 \) for B.
- The difference in their final velocities is given as \( v = v_1 - v_2 \).

2. **Displacement Equation**:
- Since both A and B cover the same distance \( S \), we can use the second equation of motion:
\[ S = \frac{1}{2} a t^2 \]
- For A:
\[ S = \frac{1}{2} a_1 t_1^2 \]
- For B:
\[ S = \frac{1}{2} a_2 t_2^2 \]

3. **Setting the Displacements Equal**:
- Since the distances are equal:
\[ \frac{1}{2} a_1 t_1^2 = \frac{1}{2} a_2 t_2^2 \]
- This simplifies to:
\[ a_1 t_1^2 = a_2 t_2^2 \]

4. **Relating Velocities and Times**:
- The final velocity can be expressed as:
\[ v_1 = a_1 t_1 \quad \text{and} \quad v_2 = a_2 t_2 \]
- From the displacement equations, we can express \( t_1 \) and \( t_2 \) in terms of \( S \):
\[ t_1 = \sqrt{\frac{2S}{a_1}} \quad \text{and} \quad t_2 = \sqrt{\frac{2S}{a_2}} \]

5. **Finding the Ratio of Velocities**:
- The ratio of the velocities can be expressed as:
\[ \frac{v_1}{v_2} = \frac{a_1 t_1}{a_2 t_2} \]
- Substituting for \( t_1 \) and \( t_2 \):
\[ \frac{v_1}{v_2} = \frac{a_1 \sqrt{\frac{2S}{a_1}}}{a_2 \sqrt{\frac{2S}{a_2}}} = \frac{a_1 \sqrt{a_2}}{a_2 \sqrt{a_1}} = \sqrt{\frac{a_1}{a_2}} \]

6. **Expressing the Velocity Difference**:
- We know from the problem statement that:
\[ v = v_1 - v_2 \]
- Substituting \( v_1 \) and \( v_2 \):
\[ v = a_1 t_1 - a_2 t_2 \]
- Using the time difference \( t_2 - t_1 = t \):
\[ v = a_1 t_1 - a_2 (t_1 + t) \]
- Simplifying gives:
\[ v = a_1 t_1 - a_2 t_1 - a_2 t = (a_1 - a_2) t_1 - a_2 t \]

7. **Final Expression**:
- Rearranging gives us the final expression for the difference in velocities:
\[ v = \sqrt{2 a_2 S} \cdot \frac{t}{\sqrt{2S/a_1}} = \sqrt{2 a_2 a_1} t \]

### Conclusion:
The value of \( v \) is given by:
\[ v = \sqrt{2 a_2 a_1} t \]