A passenger is standing `d` metres away from a bus. The bus begins to move eith constat acceleration `a. To catch the bus the passenger runs at a constant speed (v) towards the bus, at what minimum speed he must have ,so that he may catch the bus.

Explanation

Let the passenger catch the bus after time t.
The distance travelled by the bus, `" "s_(1)=0+1/2 at^(2)` …(i)
and the distance travelled by the passenger `" "s_(2)=ut+0` …(ii)
Now the passenger will catch the bus if `d+s_(1)=s_(2)` ...(iii)
`rArr d+1/2 at^(2)=ut rArr 1/2 at^(2)-ut+d=0rArr t=([u+-sqrt(u^(2)-2ad)])/a`
so the passenger will catch the bus if t is real, i.e., `u^(2) ge 2ad rArr u^(2) ge u^(2)sqrt(2ad)`
So the minimum speed of passenger for catching the bus is `sqrt(2ad)`.