Question

When acceleration be function of velocity as $a=f\left(v\right)$, then

• A. The displacement $\left(x\right)=\int \frac{vdv}{f\left(v\right)}$
• B. The acceleration may be constant
• C. The slope of acceleration versus velocity graph may be constant.
• D. (a) and (c) are correct.

Answer 1

The correct option is D (a) and (c) are correct.

It is given that $a=f\left(v\right)$
$\Rightarrow v\frac{dv}{dx}=f\left(v\right)$
$\Rightarrow \underset{0}{\overset{x}{\int }}dx=\int \frac{vdv}{f\left(v\right)}$
$\Rightarrow x=\int \frac{vdv}{f\left(v\right)}$
If acceleration is a linear function of velocity, the $a-v$ graph will be a straight line, e.g., if $a=4v+5$ and $a-v$ graph will be straight line. Hence, in this case slope of $a-v$ graph will be constant.
As acceleration is dependent on velocity and acceleration will change $v$, so acceleration will also change and can’t remain constant.
So, the correct answer is option (d).