Question

26' A vehicle starting its motion from rest travels at a uniform acceleration of 5ms-2 for a certain time and

travels with the attained uniform velocity for another certain period of time. Then it decelerates uniformly at

5ms-2 and stops after a certain time. If the total time spent is 25s and the average velocity for the whole

motion is 20ms-1 , find the time that the vehicle travelled at uniform velocity.

Answer 1

Given:

$v_0=0$

$a_1=\rm 5\: m/s^2$

$t=25\:\rm s$

$v_{\rm ave}=20\:\rm m/s$

Let us denote $\tau$ the time that the vehicle travelled at uniform velocity. Then, the total distance rtaveled by vehicle

$$d=a((25-\tau)/2)^2+a(25-\tau)/2\tau$$

The average velocity

$$v_{\rm ave}=\frac{d}{t}=\frac{a((25-\tau)/2)^2+a(25-\tau)/2\tau}{t}$$

We get an equation

$$\frac{5((25-\tau)/2)^2+5(25-\tau)/2\tau}{25}=20$$

$$((25-\tau)/2)^2+\tau(25-\tau)/2=100$$

$$(25-\tau)^2+2\tau(25-\tau)=400$$

$$\tau^2=75$$

$$\tau=5\sqrt{3}\:\rm s$$