Question

A particle is resting on a smooth horizontal floor. At $t=0$, a horizontal force starts acting on it. The magnitude of the force increases with time according to law $F=\alpha t$, where $\alpha$ is a constant. For the figure shown which of the following statement is wrong?

• A. $\text{Curve 1 shows acceleraion against time}$
• B. $\text{Curve 2 shows velocity against time}$
• C. $\text{Curve 2 shows velocity against acceleration}$
• D. $\text{Curve 1 shows acceleration against velocity}$

Answer 1

The correct option is D $\text{Curve 1 shows acceleration against velocity}$

We have from the given condition,
$F=\alpha t$
$\Rightarrow ma=\alpha t$
$\Rightarrow a=\frac{\alpha t}{m}\Rightarrow a\propto t$ ... (1) St.line
Also, $a=\frac{dv}{dt}=\frac{\alpha t}{m}\Rightarrow v=\frac{\alpha }{m}\frac{t^2}{2}$ (On Integration)
$\Rightarrow v\propto t^2$ ... (2) Parabola
Putting the value of $t$ from eq (1) into eq (2), we get
$v\propto a^2$ Parabola along acceleration axis.

So, the curve $1$ does not represent the relation between acceleration and velocity.

i.e. option 'd' is the right answer.