Question

The displacement of a particle moving in a straight line depends on time as
$x=\alpha t^3+\beta t^2+\gamma t+\delta$
The ratio of initial acceleration to its initial velocity depends :

• A. only on $\alpha$ and $\beta$
• B. only on $\beta$ and $\gamma$
• C. only on $\alpha$ and $\beta$
• D. only on $\alpha$

Answer 1

Answer: (B)

Rate of change of displacement is velocity and rate of change of velocity is acceleration.
The given equation for displacement is
$x=\alpha t^3+\beta t^2+\gamma t+\delta$
We know that velocity $=\frac{dx}{dt}$
$\therefore v=\frac{dx}{dt}=3t^2\alpha +2t\beta +\gamma$
Initial velocity when $t=0$, is
$v_0=3\alpha \times 0+2\beta \times 0+\gamma =\gamma$ ...(i)
Also acceleration $=\frac{dv}{dt}$
$\therefore a=\frac{dv}{dt}=\frac{d^{2}x}{d{t}^2}=6t\alpha +2\beta$
Initial acceleration when $t=0$, is
$a_0=6t\times 0+2\beta =2\beta$ ...(ii)
Ratio of initial acceleration to initial velocity is
$\frac{a_0}{v_0}=\frac{2\beta }{\gamma }$
which shown that this ratio depends only on $\beta$ and $\gamma$.