Question

The position of a particle at time $ t $ is given by $ \vec{r} = \vec{r_0}(1 + at) $, where $ \vec{r_0} $ and $ a $ are two constants. When will the particle come back to its starting position?

• A. \( \frac{1}{a^2} \)
• B. \( \frac{1}{a} \)
• C. \( a \)
• D. \( a^2 \)

Hint:

For problems involving linear motion with time-dependent position, solve for the time when the position becomes zero to determine when the object returns to its starting point.

Answer 1

The Correct Option is B

Understanding the equation of motion.
The position of the particle is
given by:
\[ \vec{r} = \vec{r_0}(1 + at) \] For the particle to return to its starting position, the displacement \( \vec{r} \) must be zero. Thus, we set \( \vec{r} = 0 \): \[ 0 = \vec{r_0}(1 + at) \] Solving for \( t \), we get: \[ 1 + at = 0 \quad \Rightarrow \quad t = -\frac{1}{a} \] Thus, the particle returns to its starting position after \( t = \frac{1}{a} \). Thus, the correct answer is
(B) \( \frac{1}{a} \)
.