**Problem Analysis**

Let's analyze the given problem step by step:

- Akshat starts moving with a velocity of 20 m/s.
- Hariom starts moving after 2 seconds.
- Both of them have to reach a common destination of 1000 meters.

We need to find the initial velocity of Hariom, given that the ratio of their acceleration is 2:3.

**Solution**

To solve the problem, we can use the equations of motion. Let's break down the solution into steps:

**Step 1: Determine the time taken by Akshat to reach the destination**

We can use the equation of motion:

\[s = ut + \frac{1}{2}at^2\]

where:
- s = distance traveled (1000 m)
- u = initial velocity (20 m/s)
- a = acceleration (to be determined)
- t = time taken (to be determined)

Substituting the given values into the equation, we get:

\[1000 = 20t + \frac{1}{2}at^2\]

**Step 2: Determine the time taken by Hariom to reach the destination**

Hariom starts moving 2 seconds after Akshat. Therefore, the time taken by Hariom would be t - 2 seconds.

**Step 3: Determine the acceleration of Hariom**

Given that the ratio of their acceleration is 2:3, we can write:

\[\frac{a_{\text{Hariom}}}{a_{\text{Akshat}}} = \frac{2}{3}\]

Simplifying the equation, we get:

\[a_{\text{Hariom}} = \frac{2}{3} \cdot a_{\text{Akshat}}\]

**Step 4: Determine the initial velocity of Hariom**

To find the initial velocity of Hariom, we need to use the equation of motion:

\[s = ut + \frac{1}{2}at^2\]

where:
- s = distance traveled (1000 m)
- u = initial velocity (to be determined)
- a = acceleration (\(a_{\text{Hariom}}\))
- t = time taken (to be determined)

Substituting the values into the equation, we get:

\[1000 = u(t - 2) + \frac{1}{2} \cdot \left(\frac{2}{3} \cdot a_{\text{Akshat}}\right) \cdot (t - 2)^2\]

Simplifying the equation, we get:

\[1000 = ut - 2u + \frac{1}{3} \cdot a_{\text{Akshat}} \cdot (t^2 - 4t + 4)\]

**Step 5: Substitute the values and solve for u**

Substituting the known values into the equation, we get:

\[1000 = ut - 2u + \frac{1}{3} \cdot a_{\text{Akshat}} \cdot (t^2 - 4t + 4)\]

Now, we can solve this equation to find the value of u.

**Conclusion**

In conclusion, to find the initial velocity of Hariom, we need to solve the equation obtained in Step 5. The value of u will depend