What is the minimum value of c for which the man will catch the bus?

i found out that:





x(man)t(catch)=x(bus)t(catch)





and





-b+ct(catch)=(1/2)at^2(catch)





A man is running at speed c (much less than the speed of light) to catch a bus already at a stop. At t=0 , when he is a distance b from the door to the bus, the bus starts moving with the positive acceleration a .


Use a coordinate system with x=0 at the door of the stopped bus.





So,





What is the minimum value of c for which the man will catch the bus?





Express the minimum value for the man's speed in terms of a and b ?





PLEASE clear explanations...thanks

What is the minimum value of c for which the man will catch the bus?
Let's start with the man having constant velocity





x(t)=-b+c*t


the bus has


x(t)=.5*a*t^2


when the x(t)'s are equal, we have a catch


-b+c*t=.5*a*t^2





this is a quadratic, if there is a catch, there will be a real, positive root





0=.5*a*t^2-c*t+b


the roots are


(c^2+/-sqrt(c^2-2*a*b))/a





the smaller root will be


(c^2-sqrt(c^2-2*a*b))/a





The larger root occurs if the man runs past the bus and the bus accelerates back to him.





This requires that


(c^2-2*a*b)%26gt;=0





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