Question

The law of motion of a body moving along a straight line is $x=\frac{1}{2}vt,x$ being its distance from a fixed point on the line at time $t$ and $v$ is its velocity there. Then

• A. acceleration $f$ varies directly with $x$
• B. acceleration $f$ varies inversely with $x$
• C. acceleration $f$ is constant
• D. acceleration $f$ varies directly with $t$

Answer 1

The correct option is C acceleration $f$ is constant

$x=\frac{1}{2}vt\Rightarrow x=\frac{1}{2}.\frac{dx}{dt}.t\Rightarrow 2\frac{dt}{t}=\frac{dx}{x}$
Integrating
$\Rightarrow \log x=\log t^2+\log c\Rightarrow x=t^{2}c\cdots \left(1\right)$
We know that acceleration
$f=\frac{d^{2}x}{d{t}^2}\Rightarrow f=\frac{d}{dt}\left(2tc\right)\Rightarrow f=2c$
Hence acceleration is constant