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

Options

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

Explanation

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

### Step 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:
\[ F = \frac{dp}{dt} \]

### Step 2: Differentiate the momentum function
Given the momentum function:
\[ p = 2 + 3t^2 \]
we need to differentiate \( p \) with respect to \( t \) to find \( \frac{dp}{dt} \).

Differentiating \( p \):
\[ \frac{dp}{dt} = \frac{d}{dt}(2 + 3t^2) = 0 + 6t = 6t \]

### Step 3: Substitute \( t = 3 \) seconds into the derivative
Now, we will substitute \( t = 3 \) seconds into the expression for \( \frac{dp}{dt} \):
\[ F = 6t \quad \text{at } t = 3 \]
\[ F = 6 \times 3 = 18 \, \text{N} \]

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

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