The Problem:
A body is thrown vertically upward with an initial velocity of 52 m/s from the ground. It passes twice a point at a certain height above the ground, with an interval of 10 seconds between the two passes. We need to determine the height, h, at which the body passes this point.

Given:
- Initial velocity, u = 52 m/s (upward)
- Time interval between two passes, t = 10 s
- Acceleration due to gravity, g = 10 m/s²

Approach:
To solve this problem, we can use the equations of motion for an object in free fall. We will consider the upward motion and downward motion separately.

1. Upward Motion:
During the upward motion, the body decelerates due to the gravitational force until it reaches its highest point. The final velocity (v) at this point will be zero. We can use the following equation of motion to find the height reached by the body during this phase:

v² = u² - 2gh

where:
- v = final velocity (0 m/s)
- u = initial velocity (52 m/s)
- g = acceleration due to gravity (10 m/s²)
- h = height reached

Substituting the given values into the equation, we can solve for h:

0² = (52)² - 2(10)h

0 = 2704 - 20h

20h = 2704

h = 2704/20


After reaching the highest point, the body starts falling downwards. During this phase, the body will pass the point at height h twice - once during the upward journey and once during the downward journey. The time interval between these two passes is given as 10 seconds.

We know that the time taken to reach the highest point during the upward motion is given by:

t = (v - u)/g

where:
- t = time taken (10 s)
- v = final velocity (0 m/s)
- u = initial velocity (52 m/s)
- g = acceleration due to gravity (10 m/s²)

Substituting the given values into the equation, we can solve for v (the velocity at the highest point):

10 = (0 - 52)/(-10)

10 = (-52)/(-10)

10 = 5.2

Since the time taken to reach the highest point is 5.2 seconds, the total time taken for the entire motion (upward and downward) is 10 + 5.2 = 15.2 seconds.


Now that we know the total time taken for the motion is 15.2 seconds, we can find the height h at which the body passes the point twice.

During the downward motion, the body covers the same distance h in 10 seconds. Therefore, the average velocity during this phase can be calculated as:

Average velocity = (Change in distance)/(Change in time) = h/10

Since the total time for the motion is 15.2 seconds, the time taken for the upward motion is 15.2 - 10 = 5.2 seconds. During this time, the body covers the same distance h. Therefore, the average