Problem: Find the maximum height attained by a particle thrown vertically upwards if its velocity at half of the maximum height is 10m/s. (g=10m/s^2)

Solution:
To solve this problem, we need to use the equations of motion for a particle in free fall. There are three equations of motion, but we will use the one that relates the final velocity, initial velocity, acceleration, and displacement. This equation is:

v^2 = u^2 + 2as

where v is the final velocity, u is the initial velocity, a is the acceleration (in this case, gravity), and s is the displacement.

We can use this equation twice, once for the particle's motion from the ground to half of the maximum height, and again for its motion from half of the maximum height to the maximum height. We will then add the two displacements together to find the maximum height.

Step 1: Find the velocity at the maximum height

At half of the maximum height, the particle's velocity is 10m/s. We can use this as the initial velocity for the second part of the journey (from half of the maximum height to the maximum height).

Using the equation of motion, we can find the final velocity at the maximum height:

v^2 = u^2 + 2as
0 = 10^2 - 2(10)(s)
s = 5m/s^2

Therefore, the final velocity at the maximum height is 0m/s.

Step 2: Find the displacement for each part of the journey

Using the same equation of motion, we can find the displacement for each part of the journey:

For the first part of the journey (from the ground to half of the maximum height):

v^2 = u^2 + 2as
0 = u^2 + 2(-10)(s/2)
s/2 = u^2/20

For the second part of the journey (from half of the maximum height to the maximum height):

v^2 = u^2 + 2as
0 = 10^2 + 2(-10)(s/2)
s/2 = 50

Step 3: Find the maximum height

To find the maximum height, we add the two displacements together:

s = s/2 + s/2
s = u^2/20 + 50
s = (10^2)/20 + 50
s = 55m

Therefore, the maximum height attained by the particle is 55m.