Question

The displacement-time equation for a particle in linear motion is given as $x=\frac{a}{b}(1-e^{-bt)})$. Which of the following options is/are correct?

• A. The velocity and acceleration of the particle at $t=0$ is $a$ and $-ab$ respectively.
• B. The velocity decreases as the time increases.
• C. The displacement of the particle lies between
  $$\begin{array}{c}0\le x\le \frac{a}{b}\end{array}$$
• D. The maximum magnitude of acceleration during motion is $ab$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A The velocity and acceleration of the particle at $t=0$ is $a$ and $-ab$ respectively.
B The velocity decreases as the time increases.
C The displacement of the particle lies between
$$\begin{array}{c}0\le x\le \frac{a}{b}\end{array}$$
D The maximum magnitude of acceleration during motion is $ab$

$$\begin{array}{c}x=\frac{a}{b}(1-e^{-bt})\\ \text{Velocity}\\ v=\frac{dx}{dt}=a{e}^{-bt}\\ \text{Acceleration}\\ A=\frac{dv}{dt}=-ab{e}^{-bt}\\ \text{So, at}t=0\\ v=a\text{and}A=-ab\\ \text{Also}\text{with time, velocity decreases.}\\ \text{magnitude of maximum acceleration is}ab\text{.}\\ \text{Now at}t=0\\ x=0\\ \text{when}t\to \infty \\ x=\frac{a}{b}\\ \text{So,}\\ 0\le x\le \frac{a}{b}\\ \text{All the four options are correct.}\end{array}$$