Question

The linear momentum of a particle as a function of time is given as $p = (3t^2 + 2t + 1)$ kgms$^{-1}$. Then, the force acting on the particle at $t = 3s$ will be}

• A. 20 N
• B. 10 N
• C. 15 N
• D. 2 N
• E. 8 N

Hint:

Whenever a physical quantity is given as a function of time, its "rate of change" (like force from momentum or velocity from displacement) is always found by differentiation.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
According to Newton's Second Law of Motion, the force acting on a particle is equal to the rate of change of its linear momentum with respect to time.
Mathematically, \( F = \frac{dp}{dt} \).
Key Formula or Approach:
1. Differentiate the momentum function \( p(t) \) with respect to time.
2. Substitute the value of \( t \) into the resulting derivative expression.

Step 2: Detailed Explanation:
Given momentum function:
\[ p = 3t^2 + 2t + 1 \]
The instantaneous force is:
\[ F = \frac{d}{dt}(3t^2 + 2t + 1) \]
\[ F = 6t + 2 \]
At \( t = 3s \), the force is:
\[ F = 6(3) + 2 \]
\[ F = 18 + 2 = 20 \text{ N} \]

Step 3: Final Answer:
The force acting on the particle at \( t = 3s \) is 20 N.