How do I find the time taken for A to overtake B? (Details below)

Question

Two particles A and B are moving in the same direction on parallel horizontal tracks. At a certain point, A which is travelling with a speed of 7m/s and accelerating uniformly at 1.5m/s^2 overtakes B which is travelling at 3m/s and accelerating uniformly at 2.5m/s^2 .

a) Calculate the period of time which elapses before B overtakes A

b) If, after this time, B then ceases to accelerate and continues at constant speed, calculate the time taken for A to overtake B again

Answer 1

1. Write the formula for distance as a function of velocity and acceleration for A and B.
2. Ans a) is when the time "t" for A and B are equal, i.e. set the formulas as a function of "t" equal to each other and solve for "t"
3. Ans b) Write the formula for B without the acceleration term. Starting with the time "t" from ans A, find the next time "t" where A = B.

Answer 2

d(t) = at^2/2 +vot

for A d=7t^2/2 + 1.5t

for B d=3t^2/2 +2.5t

7t^2/2 +1.5t = 3t^2/2 +2.5t

4t^2/2 = t

2t^2 =t

2t =1

t= 1/2 second

then for B d=3(1/2)^2/2 +2.5t = 3/4+2.5t

for A d still = 7t^2/2 +1.5t

7t^2/2 +1.5t = 3/4 +2.5t

7t^2/2 -t -3/4 = 0

14t^2 -4t -3 = 0

t= 4/28 + (1/28)sqr(16+168) = 1/7+sqr184/28

t = about .143+.484

t= about 0.627 seconds = a little less than 2/3 of a second