A body has a momentum of P initially. Force of F (in Newton) in the direction of momentum acts on it for T seconds after that another force of magnitude 3F opposite to the direction of momentum P acts on it for 3T seconds Now the magnitude of momentum of the body is found to be 3p Now another force F acts on it and brings it to rest The duration in seconds for which this force shall act is

A) 2t B) 6t C) 4t D) 8t

To find the answer, we need to use the principle of conservation of momentum. According to this principle, the total momentum of an isolated system remains constant if no external forces are acting on it.

Let's break down the problem step by step:

Step 1: Initial momentum (P)
The body has an initial momentum of P.

Step 2: First force (F) acting for time (T)
A force of magnitude F acts in the direction of momentum for time T. This force will change the momentum of the body.

By using the equation:

Change in momentum = Force × Time,
ΔP = F × T

The final momentum after this force acts on the body is given as:

Momentum after first force = P + ΔP = P + F × T

Step 3: Second force (3F) acting for time (3T)
Another force of magnitude 3F acts opposite to the direction of momentum for time 3T. This force will change the momentum of the body further.

Similar to Step 2, the change in momentum due to the second force is given by:

ΔP' = (3F) × (3T) = 9F × T

The final momentum after the second force acts on the body is given as:

Momentum after second force = (P + F × T) - ΔP' = (P + F × T) - (9F × T)

We are given that the magnitude of momentum after the second force is 3P:

| Momentum after second force | = 3P

So, | P + F × T - 9F × T | = 3P

Step 4: Third force (F) bringing the body to rest
Finally, another force F acts on the body to bring it to rest. We need to find the duration for which this force acts (let's call it X).

When the body comes to rest, its final momentum is zero:

Momentum after third force = 0

Using the equation:

Change in momentum = Force × Time,
ΔP'' = F × X

The final momentum after the third force acts on the body is given as:

Momentum after third force = (P + F × T - 9F × T) - ΔP'' = (P + F × T - 9F × T) - (F × X)

Since the body comes to rest, the final momentum is zero:

(P + F × T - 9F × T) - (F × X) = 0

Simplifying the equation:

P + F × T - 9F × T - F × X = 0

Rearranging the terms:

P - F × X = 9F × T - F × T

P - F × X = 8F × T

Therefore, the duration for which the third force should act (X) is given by:

X = (P - 8F × T) / F

Now we can analyze the answer choices to find the correct option.

The correct answer is D) 8t, based on the equation we derived.

Hope this explanation helps!