Question

An object moving with an speed of $25\text{m/s}$ is decelerating at a rate give by $a=-5\sqrt{v},$ where, $v$ is speed at any time. The time taken by the object to come at rest is

• A. $1\text{s}$
• B. $5\text{s}$
• C. $2\text{s}$
• D. $3\text{s}$

Answer 1

The correct option is C $2\text{s}$

As per the defination of acceleration , we can write
$\frac{dv}{dt}=-5\sqrt{v}$
$\Rightarrow \frac{dv}{\sqrt{v}}=-5dt$
On integrating above equation, we get
$\Rightarrow \int \frac{dv}{\sqrt{v}}=\int (-5)dt$
$2\sqrt{v}=-5t+c$ ..........(1)
Where $c$ is integrating constant.
To evaluate the constant just put $t=0$ and $v=25$ as given in the question.
By solving for $c$ we get $c=10$.
Now on putting $c=10$ in equestion (1) we get
$2\sqrt{v}=-5t+10$ ..............(2)
Now put $v=0$ in equation (2) to get the desired time.
$\Rightarrow 2\left(0\right)=-5t+10$
$\Rightarrow 5t=10$
$\Rightarrow t=2\text{s}$