Question

A car starts moving along a line first with acceleration a=5ms-2 starting from rest then uniformly and finally decelerating at same rate a and comes to rest.The total time of motion is 25s.The average speed during time is 20 ms-1. How long does particle move uniformly?

Answer 1

The average speed can be calculated as total distance over total time:

$$s_\text{av}=\frac{D}{T}.$$

The total time is the sum of time when the car accelerated, moved uniformly with speed $s_u$ within the distance $d_u$, and decelerated:

$$T=\frac{s_u}{a}+\frac{d_u}{s_u}+\frac{s_u}{a}=\frac{2s_u}{a}+\frac{d_u}{s_u}.$$

The total distance the car traveled is:

$$D=s_\text{av}T=\frac{s_u^2}{2a}+d_u+\frac{s_u^2}{2a}=\frac{s_u^2}{a}+d_u.$$

So, using the first expression, we have two equations to solve them for the speed and distance labeled with index u:

$$500=\frac{s_u^2}{5}+d_u,$$

$$25=\frac{2s_u}{5}+\frac{d_u}{s_u}.$$

This will give you a quadratic equation for speed and the following pair of roots is the solution:

$$s_u=25\text{ m/s},\space d_u=375\text{ m},\\ s_u=100\text{ m/s},\space d_u=-1500\text{ m}.$$

Both roots give the same time of uniform motion:

$$t_u=\frac{d_u}{s_u}=15\text{ s}.$$