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 solve the problem, we need to find the force acting on the particle at \( t = 3 \) seconds, given the momentum \( P \) as a function of time \( t \).

### Step-by-Step Solution:

1. **Understand the relationship between momentum and force**:
The force \( F \) acting on an object is related to the rate of change of momentum \( P \) with respect to time \( t \). Mathematically, this is expressed as:
\[ F = \frac{dP}{dt} \]

2. **Given momentum function**:
We are given the momentum of the particle as:
\[ P = 2 + 3t^2 \]

3. **Differentiate the momentum with respect to time**:
To find the force, we need to differentiate the momentum function \( P \) with respect to \( t \):
\[ \frac{dP}{dt} = \frac{d}{dt}(2 + 3t^2) \]
The derivative of a constant (2) is 0, and the derivative of \( 3t^2 \) is \( 6t \):
\[ \frac{dP}{dt} = 6t \]

4. **Substitute \( t = 3 \) seconds into the derivative**:
Now we need to find the force at \( t = 3 \) seconds:
\[ F = 6t \quad \text{at} \quad t = 3 \]
\[ F = 6 \times 3 = 18 \, \text{N} \]

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

### Summary:
The force acting on the particle at \( t = 3 \) seconds is \( 18 \, \text{N} \).