A particle starts with an initial velocity and passes successively over the two halves of a given distance with constant accelerations `a_1 and a_2` respectively. Show that the final velocity is the same as if the whole distance is covered with a uniform acceleration `(a_1 + a_2)/2 .`

Explanation

To solve the problem, we need to analyze the motion of the particle in two segments, each with different accelerations, and then show that the final velocity can also be expressed using a uniform acceleration over the entire distance.

### Step-by-Step Solution:

1. **Define the Variables:**
- Let the initial velocity of the particle be \( u \).
- The total distance traveled by the particle is \( 2s \) (where \( s \) is the distance for each half).
- The first half of the distance \( s \) is traveled with acceleration \( a_1 \).
- The second half of the distance \( s \) is traveled with acceleration \( a_2 \).

2. **Equation for the First Half:**
- For the first half of the distance, we can use the kinematic equation:
\[ v_1^2 = u^2 + 2a_1s \]
- Here, \( v_1 \) is the final velocity after traveling the first half.

3. **Equation for the Second Half:**
- The initial velocity for the second half is \( v_1 \). The kinematic equation for the second half is:
\[ v_2^2 = v_1^2 + 2a_2s \]
- Here, \( v_2 \) is the final velocity after traveling the second half.

4. **Substituting \( v_1^2 \) into the Second Equation:**
- Substitute the expression for \( v_1^2 \) from the first equation into the second equation:
\[ v_2^2 = (u^2 + 2a_1s) + 2a_2s \]
- Simplifying this gives:
\[ v_2^2 = u^2 + 2a_1s + 2a_2s = u^2 + 2(a_1 + a_2)s \]

5. **Combine the Two Segments:**
- The total distance is \( 2s \). If we consider the entire distance covered with a uniform acceleration \( a \), where \( a = \frac{a_1 + a_2}{2} \), we can use the kinematic equation:
\[ v^2 = u^2 + 2as \]
- Substituting \( a \) and the total distance \( 2s \):
\[ v^2 = u^2 + 2\left(\frac{a_1 + a_2}{2}\right)(2s) \]
- This simplifies to:
\[ v^2 = u^2 + 2(a_1 + a_2)s \]

6. **Conclusion:**
- From both approaches, we have:
\[ v_2^2 = u^2 + 2(a_1 + a_2)s \]
\[ v^2 = u^2 + 2(a_1 + a_2)s \]
- Therefore, we conclude that:
\[ v_2 = v \]
- This shows that the final velocity \( v_2 \) is the same as if the whole distance is covered with uniform acceleration \( \frac{a_1 + a_2}{2} \).