Question

When acceleration is a function of velocity as $$a=f(v)$$. Then:
A) The displacement $$\Rightarrow x=\int {\displaystyle \frac{vdv}{f(v)}}$$
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

Hint: Clearly as the options are framed in such a manner that we must calculate displacement. Plot graph of acceleration versus velocity and check whether it is constant or not. Acceleration must also be calculated. Basic knowledge of integration and differentiation is needed. Recall the relation between displacement, velocity and acceleration in calculus forms.

Complete step by step solution:
Since the acceleration is given as a function of velocity and options deal with displacement and velocity therefore we have to calculate velocity and displacement from the given equation. It is known that the derivative of displacement is velocity and the derivative of velocity with respect to time is acceleration.
The given equation is: $$a=f(v)$$
As we also know that $$a=v{\displaystyle \frac{dv}{dx}}$$ therefore we have
$$\Rightarrow v{\displaystyle \frac{dv}{dx}}=f(v)$$
$$\Rightarrow dx=\int {\displaystyle \frac{vdv}{f(v)}}$$

Hence Option (A) is correct.

If the value of $$f(v)=k$$
Where $$k$$ is a constant.
Then the acceleration for the given equation will also be constant.

Hence Option (B) can also be correct.

The given equation $$a=f(v)$$ is of the form of an equation of a straight line with c-intercept being zero. Now $$f(v)$$ will be a linear function of velocity then the graph of acceleration versus time will be a straight line and hence the slope will be constant. Since the slope of any straight line is constant.

Hence Option (C) can also be correct.

Note: We are not aware whether the function $$f(v)$$ is a constant function or a linear function. Hence we cannot say for sure what the correct option is. Also it can be said that option A is correct in any case if the function's velocity is non-zero.