Question

A car starts moving along a line, first with acceleration $a=5\frac{m}{s^2}$ starting from rest, then uniformly, and finally decelerating at the same rate $a$ to come to rest. The total time of motion is $25s$. The average speed during the time is $20\frac{m}{s}.$ How long does the particle move uniformly?

Answer 1

$v-t$ graph for given situation can be considered as


Let $t$ be time of accelerated motion,
$v$ be velocity during uniform motion
here $v=5t$, as final velocity of accelerated motion is velocity during uniform motion.

Total time is given as $25s$, so $2t+x=25\Rightarrow x=25-2t$
Total displacement is area under $v-t$ graph$=\frac{1}{2}\times 5t\times t+\left(5t\right)(25-2t)+\frac{1}{2}\times 5t\times t$

$\text{Average velocity}=\frac{\text{Total displacement}}{\text{Total time}}$

$20=\frac{\frac{1}{2}\times 5t\times t+\left(5t\right)(25-2t)+\frac{1}{2}\times 5t\times t}{25}$
$500=5t^2-10t^2+125t$
$5t^2-125t+500=0$
$t^2-25t+100=0$
$t=5$
Thus, the particle moved uniformly for $25-2t=15s$.