A car, starting from rest, is accelerated at a constant rate α until it attains a speed ν. It is then retarded at a constant rate β until it comes to rest. The average speed of the car during its entire journey is

The distance s1 covered by the car during the time it is accelerated is given by 2αS1 = v2, which gives s1=v2/2α. The distance s2 covered during the time the car is decelerated is, similarly given by s2=v2/2β. Therefore, the total distance covered is s=s1+s2=v221α+1β  ...(i)If t1 is the time of acceleration and t2 that of deceleration, then v=αt1=βt2 or t1=v/α and t2=v/βt=t1+t2=v1α+1β  ...(ii)From (i) and (ii), the average speed of the car is given by total distancetotal time=st=v2.