Inserting the formulas you found for

x_(man)(t)

and

x_(bus )(t)

into the condition

x_(man)(t_(catch ))=x_(bus )(t_(catch ))

, you obtain the following:\

-d+v_(man )t_(catch )=(1)/(2)a_(bus )t_(catch )^(2) or (1)/(2)a_(bus )t_(catch )^(2)-v_(man )t_(catch )+d=0

\ Intuitively, the man will not catch the bus unless he is running fast enough. In mathematical terms, there is a constraint on the man's speed

v_(man)

so that the equation above gives a solution for

t_(catch )

that is a real positive number.\ Find

v_(man,min)

, the minimum value of

v_(man)

for which the man will catch the bus.\ Express the minimum value for the man's speed in terms of

a_(bus )

and

d

.

Inserting the formulas you found for xman(t) and xbus(t) into the condition xman(tcatch)=xbus(tcatch), you obtain the following: d+vmantcatch=21abustcatch2,or21abustcatch2vmantcatch+d=0. Intuitively, the man will not catch the bus unless he is running fast enough. In mathematical terms, there is a constraint on the man's speed vman so that the equation above gives a solution for tcatch that is a real positive number. Find vman,min, the minimum value of vman for which the man will catch the bus. Express the minimum value for the man's speed in terms of abus and d