Answer:

5u/8

Explanation:

u =velocity of P

U=velocity of Q

h= height of P

H= height of Q

we know that,

h/H=64/25

u^2. ×. 2g. =. 64

2g. U^2. 25

from this we get ,

U=5u

8

Hope it helps

**Given:**

Particle P is projected vertically upwards from point A.

Six seconds later, another particle Q is projected vertically upward from A.

Both P and Q reach a maximum height simultaneously.

The ratio of their maximum heights reached by P and Q is 64:25.

**To find:**

Velocity of projection of Q.

**Solution:**

Let's assume the initial velocity of particle P as u and the initial velocity of particle Q as v.

**Calculating the time taken by particle P to reach maximum height:**

The time taken by particle P to reach maximum height can be calculated using the formula:

t = (v-u) / g

Where,
v = final velocity = 0 (as the particle reaches maximum height, its velocity becomes 0)
u = initial velocity of P
g = acceleration due to gravity = 9.8 m/s^2

As per the given information, particle P reaches maximum height 6 seconds after it is projected. So,

6 = (0 - u) / 9.8

Simplifying the equation, we get:

-6 * 9.8 = -u

u = 58.8 m/s

Therefore, the initial velocity of particle P is 58.8 m/s.

**Calculating the time taken by particle Q to reach maximum height:**

Particle Q is projected 6 seconds after particle P. So, the time taken by particle Q to reach maximum height will be 6 seconds less than the time taken by particle P. Let's call this time t1.

t1 = t - 6

**Calculating the maximum heights reached by particles P and Q:**

The maximum height reached by a particle can be calculated using the formula:

h = (v^2 - u^2) / (2g)

For particle P:

hP = (0^2 - u^2) / (2 * 9.8) = u^2 / (2 * 9.8)

For particle Q:

hQ = (0^2 - v^2) / (2 * 9.8) = v^2 / (2 * 9.8)

Given that the ratio of hP to hQ is 64:25, we have:

hP / hQ = 64 / 25

(u^2 / (2 * 9.8)) / (v^2 / (2 * 9.8)) = 64 / 25

Simplifying the equation, we get:

(u^2 / v^2) = 64 / 25

(u / v)^2 = 64 / 25

Taking the square root of both sides, we get:

u / v = √(64 / 25)

u / v = 8 / 5

u = (8/5) * v

Substituting the value of u obtained earlier, we have:

58.8 = (8/5) * v

Simplifying the equation, we get:

v = (58.8 * 5) / 8

v = 36.75 m/s

Therefore, the velocity of projection of particle Q is 36.75 m/s.