A bus starts from rest with a constant acceleration. At the same time a car travelling with a constant velocity $50\text{m/s}$  over-takes and passes the bus. How fast is the bus travelling when they are side by side

Let t be the time after which car meets and crosses the bus. So Bus and Car travels equal distance in time t.

For Bus:

$\text{u}=0,\text{a}=5{\text{m/s}}^{\text{2}}$

${\text{s}}_{\text{B}}=\text{ut}+\frac{1}{2}\left(5\right){\text{t}}^{\text{2}}$

$=0+\frac{1}{2}\left(5\right){\text{t}}^{\text{2}}$

${\text{s}}_{\text{B}}=2.5{\text{t}}^{\text{2}}\to \left(1\right)$

For Car:

$v=50\text{m/s}$

${\text{s}}_{\text{C}}=\text{v}\times \text{t}$                                 

$=\text{50}\times \text{t}$

${\text{s}}_{\text{C}}=50\text{t}\to \left(2\right)$

From (1) and (2)

${\text{s}}_{\text{B}}={\text{s}}_{\text{C}}$

$2.5{\text{t}}^{\text{2}}=50\text{t}$                             

$2.5\text{t}=50$

$t=\frac{50}{2.5}=\frac{500}{25}=20\sec$

Velocity of bus after 20 seconds

$\text{v}=\text{u}+\text{at}$

$=\text{0}+\left(5\right)\left(20\right)$

$\text{v}=100\text{m/s}$