Problem Cars A and B are racing each other along the same st

Problem: Cars A and B are racing each other along the same straight road in the following manner: Car A has a head start and is a distance DA beyond the starting line at t=0. The starting line is at x=0. Car A travels at a constant speed vA. Car B starts at the starting line but has a better engine than Car A, and thus Car B travels at a constant speed vB, which is greater than vA.

1.) How long after Car B started the race will Car B catch up with Car A?

Express the time in terms of given quantities.

t_catch=

2.) How far from Car B\'s starting line will the cars be when Car B passes Car A?

Express your answer in terms of known quantities. (You may use t_catch as well.)

d_pass=

Please write out each step with explanation. Thank you so much

Solution

1) suppose after time t, Car B catches Car A.

DIstance traveled by A in this time, dA = vA t

by Car B:

suppose acc of car B is a then it wil accelerate until its speed becoems vB.

tcatch = t\'

then a = vB / t\'

distance traveled in t\' = (vB / t\') (t\'^2) / 2 = vB t\' / 2

distace traveled in (t -t\') time = vB (t - t\')

= vB t - vB t\'

dB = vBt - vB t\' + vB t\'/2

dB = vB t - vB t\' /2

and dA = dB

vA t = vB t - vB t\' /2

t = vB tcatch / (2(vB - vA) )

(2) d = vA t = vA vB tcatch / (2 (vB - vA))