Question

The velocity of a particle at time $t$ is given by the relation $v = 6t - \frac{t^2}{6}$. If its displacement $S$ is zero at $t = 0$, then the distance travelled in $3\ \text{sec}$ is

• A. $\frac{51}{2}\ \text{units}$
• B. $\frac{39}{2}\ \text{units}$
• C. $\frac{57}{2}\ \text{units}$
• D. $\frac{33}{2}\ \text{units}$

Hint:

Always quickly check if the velocity function goes to zero within the integration limits. If it doesn't change sign, you can safely integrate the entire block in one single step without splitting the integral into absolute value sections.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The velocity of a moving particle is given as a function of time: $v(t) = 6t - \frac{t^2}{6}$. We are given that its initial displacement is $S(0) = 0$, and we need to calculate the total distance travelled by the particle from $t = 0$ to $t = 3\ \text{seconds}$.

Step 2: Key Formula or Approach:
Velocity is defined as the rate of change of displacement: $v = \frac{dS}{dt}$. To find displacement from a velocity equation, we integrate with respect to time: $$S(t) = \int v(t) \, dt$$ First, check if the velocity changes sign (crosses zero) within the interval $t \in [0, 3]$. Setting $v = 0 \implies 6t - \frac{t^2}{6} = 0 \implies t\left(6 - \frac{t}{6}\right) = 0 \implies t = 0$ or $t = 36\ \text{seconds}$. Since the velocity remains strictly positive throughout our entire time frame $[0, 3]$, the total distance travelled is equal to the net displacement.

Step 3: Detailed Explanation:
Set up the definite integral for displacement from $t = 0$ to $t = 3$: $$S = \int_{0}^{3} \left( 6t - \frac{t^2}{6} \right) dt$$ Apply the polynomial integration power rule $\int t^n \, dt = \frac{t^{n+1}}{n+1}$: $$S = \left[ \frac{6t^2}{2} - \frac{t^3}{6 \times 3} \right]_{0}^{3} = \left[ 3t^2 - \frac{t^3}{18} \right]_{0}^{3}$$ Substitute the upper integration limit $t = 3$ into the expression: $$S = 3(3)^2 - \frac{(3)^3}{18} = 3(9) - \frac{27}{18}$$ Simplify the terms: $$S = 27 - \frac{3}{2} = \frac{54 - 3}{2} = \frac{51}{2}\ \text{units}$$

Step 4: Final Answer:
The total distance travelled by the particle is $\frac{51}{2}\ \text{units}$, which corresponds to option (A).