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.
Step 2: Key Formula or Approach:
The formula for force is \(F = \frac{dp}{dt}\).
We need to differentiate the given expression for momentum with respect to time \(t\) and then evaluate it at \(t = 3\) seconds.
Step 3: Detailed Explanation:
Given momentum:
\[ p(t) = 3t^2 + 2t + 1 \] Differentiate \(p(t)\) with respect to \(t\):
\[ F = \frac{dp}{dt} = \frac{d}{dt}(3t^2 + 2t + 1) \] \[ F = 6t + 2 \] Now, substitute \(t = 3\) s to find the force at that specific instant:
\[ F(3) = 6(3) + 2 \] \[ F(3) = 18 + 2 = 20 \text{ N} \] Step 4: Final Answer:
The force acting on the particle is 20 N.