Problem:
A particle is dropped from the top of a tower. During its motion, it covers 9/25 th part of the tower in the last one second. Find the velocity with which it will hit the ground.

Solution:

To solve this problem, we need to consider the motion of the particle in two parts: the initial free fall and the last one second.

Part 1: Initial Free Fall
During the initial free fall, the particle is subject to only gravitational acceleration. The distance covered by the particle during this time can be calculated using the equation:

d = (1/2) * g * t^2

Where:
- d is the distance covered
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
- t is the time taken to cover the distance

Since the particle is dropped from rest, the initial velocity is zero. We can rearrange the equation to solve for time:

t = sqrt(2d/g)

Part 2: Last One Second
During the last one second of motion, the particle covers 9/25 th part of the tower. Let's denote this distance as x. We can calculate the velocity during this time using the equation:

v = x / t

Where:
- v is the velocity
- x is the distance covered in the last one second
- t is the time taken to cover the distance

Substituting the value of t from Part 1 into the equation, we get:

v = x / sqrt(2d/g)

Now, we can substitute the given values and calculate the velocity:

v = (9/25) / sqrt(2d/g)

To find the velocity with which the particle will hit the ground, we need to find the total distance covered by the particle. This distance is the height of the tower. Let's denote it as h.

Total distance = h = d + x

Substituting the values, we have:

h = d + (9/25)

The particle will hit the ground when the distance covered is equal to the height of the tower. So, we can equate h to d + (9/25):

h = d + (9/25) = d

Simplifying the equation, we get:

h = (16/25) * d

We know that the distance covered in the last one second is 9/25 th of the total distance, which is equal to (16/25) * d. Solving for d, we get:

d = (25/16) * x

Now, we can substitute the value of d in the equation for velocity:

v = (9/25) / sqrt(2 * (25/16) * x / g)

Simplifying the equation further, we get:

v = (9/