The equation of motion of a particle is $s=a t^2+b t+c$. If the displacement after 1 second is 20 m ,…

The equation of motion of a particle is $s=a t^2+b t+c$. If the displacement after 1 second is 20 m , velocity after 2 seconds is $30 \mathrm{~m} / \mathrm{sec}$ and the acceleration is $10 \mathrm{~m} / \mathrm{sec}^2$, then
  1. $a+c=2 b$
  2. $a+c=b$
  3. $\quad \mathrm{a}-\mathrm{c}=\mathrm{b}$
  4. $a+c=3 b$

Solution

$s=a t^2+b t+c$
Displacement after 1 second is 20 m . $\therefore \quad 20=a+b+c...(i)$ $\text { Velocity }=\mathrm{v}=\frac{\mathrm{ds}}{\mathrm{dt}}=2 \mathrm{at}+\mathrm{b}$
Velocity after 2 seconds is $30 \mathrm{~m} / \mathrm{sec}$ $\therefore \quad 30=4 a+b...(ii)$
Acceleration $=\frac{\mathrm{dv}}{\mathrm{dt}}=2 \mathrm{a}$ Acceleration is $10 \mathrm{~m} / \mathrm{sec}^2$ $\therefore \quad 10=2 a$ $\Rightarrow a=5$
From (ii), we get $b=10$ From (i), we get $\mathrm{c}=5$ $\therefore \quad a+c=b$

Asked in: MHT CET 2024 (15 May Shift 2)

Practice more Differentiation questions on Aicharya