Question

The acceleration, $a$, of a particle depends on displacement, $s$, as $a=s+5$. It is given that initially $s=0$ and $v=5m/s$. Then, the expression for velocity $v$ as a function of $s$ is :

• A. $v=s+5$
• B. $v=\sqrt{s+5}$
• C. $v=\sqrt{s^2+10s}$
• D. $v=s-5$

Answer 1

The correct option is A $v=s+5$

Here, $a=s+5$

So, $\frac{dv}{dt}=s+5$

or $\frac{dv}{ds}\frac{ds}{dt}=s+5$

or $\frac{dv}{ds}v=s+5$ (as $\frac{ds}{dt}=v$)

or $vdv=(s+5)ds$

Integrating both sides, $\int vdv=\int (s+5)ds$

or $\frac{v^2}{2}=\frac{s^2}{2}+5s+C$ where C is integrating constant

When $S=0,v=5,C=25/2$

So, $v^2/2=s^2/2+5s+25/2$ or $v^2=s^2+10s+25$

or $v^2=(s+5{)}^2$

or $v=s+5$