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
2t. 6t. 4t. 8t
To find the duration for which the third force brings the body to rest, we need to analyze the given scenario.
First, let's break down the problem step by step:
Step 1: Initial Momentum (P)
The body initially has a momentum of P.
Step 2: Force F for T seconds
A force of magnitude F acts on the body for T seconds in the direction of its momentum. This changes the momentum.
Step 3: Force 3F for 3T seconds
After the first force, another force of magnitude 3F acts on the body for 3T seconds in the opposite direction of its momentum. This changes the momentum further.
Step 4: Magnitude of Momentum (3P)
After the second force, the magnitude of the body's momentum is found to be 3P.
Step 5: Rest
The third force is applied to bring the body to rest. We need to determine the duration for which this force acts.
To solve this problem, we need to apply the principle of conservation of momentum.
Conservation of momentum states that the total momentum before an event is equal to the total momentum after the event, provided no external forces act on the system.
Using this principle, we can analyze the given scenario:
Step 1: Initial Momentum (P) is conserved.
Step 2: Force F for T seconds changes the momentum.
Step 3: Force 3F for 3T seconds further changes the momentum.
Step 4: Magnitude of Momentum (3P) is the final momentum.
Since we need to bring the body to rest, the final momentum after the third force is applied should be zero.
Hence, we can write the equation:
Initial Momentum + Momentum change due to Force F + Momentum change due to Force 3F = Final Momentum
P + (F * T) + (3F * 3T) = 0
Simplifying the equation:
P + FT + 9FT = 0
P + 10FT = 0
Dividing both sides by P:
1 + 10FT / P = 0
10FT / P = -1
10FT = -P
Now we need to find the duration for which the third force shall act, so we can express it as:
T_third = x * T
where x is an unknown multiplier.
Substituting this value in the equation above:
10(x * T) = -P
10xT = -P
Therefore, x can be calculated as:
x = -P / (10T)
Since x represents the multiplier for T, the duration for which the third force acts is:
T_third = x * T = (-P / (10T)) * T = -P / 10
Now, T_third = -P / 10
Since time cannot be negative, we take the magnitude of the duration:
Magnitude of T_third = P / 10
Therefore, after analyzing the given scenario using the conservation of momentum principle, we find that the magnitude of the duration (in seconds) for which the third force shall act is P / 10.
In conclusion, the answer is T_third = P / 10.