Question

A particle is moving along $x$ - axis and its coordinate with time is given as $x=-40+5t^2$. Then:

• A. Particle will cross the origin at $t=2\sqrt{2}\text{s}$.
• B. Magnitude of velocity and acceleration are equal at $t=1\text{s}$
• C. Particle will cross the origin at $t=5\sqrt{2}\text{s}$.
• D. Magnitude of velocity and acceleration are equal at $t=3\text{s}$.

Answer 1

Answer: (A), (B)

Magnitude of velocity and acceleration are equal at $t=1\text{s}$

Given
$x=-40+5t^2$
At origin, $x=0$
$\Rightarrow 0=-40+5t^2$
$\Rightarrow t=\sqrt{\frac{40}{5}}$
$\Rightarrow t=\sqrt{8}$
$\Rightarrow t=2\sqrt{2}\text{sec}$.
So, particle will cross the origin at $t=2\sqrt{2}\text{sec}$.
Now velocity of the particle is
$\frac{dx}{dt}=\frac{d}{dx}(-40+5t^2)$
$\Rightarrow v=10t$ .....(1)
Acceleration of the particle is
$\frac{dv}{dt}=\frac{d}{dt}\left(10t\right)$
$\Rightarrow a=10$ ......(2)
From equation (1) and (2),
$|a|=|v|$
$\Rightarrow 10t=10$
$\Rightarrow t=1\text{sec}$
Hence, options $\left(a\right)$ and $\left(b\right)$ are correct.