Question

A particle is resting on a smooth horizontal floor. At $t=0$, a horizontal force starts acting on it. 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 statements is/are correct ?

• A. Curve 1 may represent acceleration against time graph
• B. Curve 2 may represent velocity against time graph
• C. Curve 2 may represent velocity against acceleration graph
• D. None of these

Answer 1

Answer: (A), (B), (C)

Curve 2 may represent velocity against acceleration graph

Given $F=\alpha t$
$ma=\alpha t$
$a=\frac{\alpha t}{m}\Rightarrow a\propto t\dots \dots \left(1\right)$
$a-t$ graph is a straight line passing through the origin.
St. line $\frac{dv}{dt}=\frac{\alpha t}{m}$
$\Rightarrow v=\frac{\alpha }{m}\frac{t^2}{2}\Rightarrow v\propto t^2\dots \dots \left(2\right)$
$v$ vs $t$ is a parabola
on solving (1) and (2)
$v\propto a^2\Rightarrow$ $v-a$ graph is parabola