Question

Acceleration time graph of a particle is shown. Work done by all the forces acting on the particle of mass m in time interval $t_1$ and $t_2$ while $a_1$, is the acceleration at time $t_1$, is given by : (at $t=0,$ particle was at rest)

• A. $\frac{m{a}_1^2}{4t_1}\left(t_2^3-t_1^3\right)$
• B. $\frac{m{a}_1^2}{8t_1^2}\left(t_2^4-t_1^4\right)$
• C. $\frac{m{a}_1^2}{4t_1^2}\left(t_2^4-t_1^4\right)$
• D. $\frac{m{a}_1}{2t_1}\left(t_2^2-t_1^2\right)$

Answer 1

Answer: (B)

$W=K_f-K_i=\frac{1}{2}m\left(v_f^2-v_i^2\right)$
Area under acceleration-time graph is change in velocity.
So, $W=\frac{1}{2}m\left[{\left(\frac{1}{2}{a}_2{t}_2\right)}^2-{\left(\frac{1}{2}{a}_1{t}_1\right)}^2\right]$
$=\frac{1}{8}m\left[\frac{t_2^4}{t_1^2}{a}_1^2-a_1^2{t}_1^2\right]$
(since, $a_2=\frac{t_2}{t_1}{a}_1$)
$=\frac{m{a}_1^2}{8t_1^2}\left(t_2^4-t_1^4\right)$