The problem states that a body is traveling with uniform acceleration along a straight line and crosses two points, A and B, with velocities of 20 meters per second and 30 meters per second, respectively. We need to determine the speed of the body at the midpoint of A and B.

To solve this problem, we can use the equations of motion for uniformly accelerated motion. The equations are:

1. v = u + at
2. s = ut + (1/2)at^2
3. v^2 = u^2 + 2as

Where:
- v is the final velocity
- u is the initial velocity
- a is the acceleration
- t is the time taken
- s is the displacement

Let's break down the problem into steps:

Step 1: Find the acceleration
Since the body is traveling with uniform acceleration, we can assume the acceleration is constant. Let's denote the acceleration as 'a'.

Step 2: Find the time taken to travel from A to B
Using equation 1, we can find the time taken to travel from A to B by substituting the given values:
30 = 20 + a*t
Simplifying the equation, we get:
10 = a*t

Step 3: Find the displacement from A to B
Using equation 2, we can find the displacement from A to B by substituting the given values:
s = 20*t + (1/2)*a*t^2

Step 4: Find the time taken to reach the midpoint of A and B
Since the body is traveling with uniform acceleration, the time taken to reach the midpoint is half the time taken to travel from A to B. Let's denote this time as 't/2'.

Step 5: Find the displacement from A to the midpoint
Using equation 2, we can find the displacement from A to the midpoint by substituting the values:
s = 20*(t/2) + (1/2)*a*(t/2)^2

Step 6: Find the speed at the midpoint
Using equation 3, we can find the speed at the midpoint by substituting the values:
v^2 = 20^2 + 2*a*s

By solving these equations, we can determine the speed of the body at the midpoint of A and B.

Note: Remember to use appropriate units throughout the calculations and consider the direction of motion (positive or negative) when substituting values into the equations.