The momentum P ( in kg `ms^(-1)`) of a particle is varying with time t ( in second ) as p=2+`3t^(2)`. The force acting on the particle at t=3s will be

Options

A. 18N
B. 54N
C. 9N
D. 15N

Explanation

To find the force acting on the particle at \( t = 3 \) seconds, we will follow these steps:

### Step 1: Write down the expression for momentum
The momentum \( P \) of the particle is given by the equation:
\[ P = 2 + 3t^2 \]

### Step 2: Differentiate the momentum with respect to time
According to Newton's second law, the force \( F \) acting on an object is the rate of change of momentum with respect to time:
\[ F = \frac{dP}{dt} \]
We need to differentiate the momentum function \( P \) with respect to \( t \):
\[ \frac{dP}{dt} = \frac{d}{dt}(2 + 3t^2) \]
Since the derivative of a constant is zero and using the power rule for differentiation:
\[ \frac{dP}{dt} = 0 + 6t = 6t \]

### Step 3: Substitute \( t = 3 \) seconds into the force equation
Now we substitute \( t = 3 \) seconds into the expression for force:
\[ F = 6t = 6 \times 3 = 18 \, \text{N} \]

### Final Answer
The force acting on the particle at \( t = 3 \) seconds is:
\[ F = 18 \, \text{N} \]

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