Question

A particle moves along a straight line such that its position is given by \( x(t) = 3t^2 - t^3 \). What is its velocity at \( t = 2 \) seconds?

• A. \( 0 \text{ m/s} \)
• B. \( 4 \text{ m/s} \)
• C. \( -4 \text{ m/s} \)
• D. \( 12 \text{ m/s} \)

Hint:

Whenever position is given as a function of time, remember: \[ \text{Velocity} = \frac{dx}{dt} \] and \[ \text{Acceleration} = \frac{d^2x}{dt^2} \] Differentiate carefully term-by-term using power rule.

Answer 1

The Correct Option is A

Step 1 : Understanding the Question:
In this kinematics problem, we study the motion of a particle along a straight path where its position is represented as a function of time. To find the particle's instantaneous velocity at a specific moment, we must analyze how its position coordinate changes dynamically over time. Instantaneous velocity describes the rate of change of displacement, which mathematically corresponds to the slope of the position-time curve at that particular instant. Thus, calculating the derivative of the position function allows us to map the precise speed and direction of the particle at any given second.
Step 2 : Key Formulas and Approach:
The key relation to solve this problem is the derivative of the position function with respect to time:
\[ v(t) = \frac{dx}{dt} \] To perform the differentiation of polynomial terms, we apply the standard calculus power rule:
\[ \frac{d}{dt}(t^n) = n t^{n-1} \] Our approach is to first find the general velocity expression by differentiating the given cubic position function, and then substitute the specified time value of 2 seconds into the derivative to calculate the final numerical velocity value.
Step 3 : Detailed Solution:

Start by writing down the given position equation: \( x(t) = 3t^2 - t^3 \).

Differentiate each term of the position equation with respect to time \( t \) using the power rule:
\[ v(t) = \frac{d}{dt}(3t^2 - t^3) = 6t - 3t^2 \]
Substitute the given time \( t = 2 \) seconds directly into our newly derived velocity function:
\[ v(2) = 6(2) - 3(2)^2 \]
Evaluate the arithmetic operations to simplify the expression step-by-step:
\[ v(2) = 12 - 3(4) = 12 - 12 = 0 \text{ m/s} \]
Step 4 : Final Answer:
The velocity of the particle at \( t = 2 \) seconds is \( 0 \text{ m/s} \), which corresponds to option (A).
\[ \boxed{0 \text{ m/s}} \]