Two cars start off to race with velocities v1 & v2 & travel in a straight line accleration a1 & a2.?

Question

If the race ends at the same time, find the length of track in terms of v1,v2 & a1,a2. ( Dynamic question)

Answer 1

s = v1t + (1/2)a1t^2 = v2t + (1/2)a2t^2
t = - v1 ± √(v1^2 +2a1s) = - v2 ± √(v2^2 +2a2s)
± √(v1^2 +2a1s) ± √(v2^2 +2a2s) = v1 - v2
(v1^2 +2a1s) ± 2√(v1^2 +2a1s)√(v2^2 +2a2s) + (v1^2 +2a2s) = v1^2 - v1v2 + v2^2
+ 2a1s ± 2√(v1^2 +2a1s)√(v2^2 +2a2s) + 2a2s) = - 2v1v2
± √(v1^2 +2a1s)√(v2^2 +2a2s) = - a1s - a2s - v1v2
± √(v1^2 +2a1s)(v2^2 +2a2s) = - (a1 + a2)s - v1v2
± √(v1^2v2^2 +2a2sv1^2 + 2a1sv2^2 + 4a1a2s^2) = - ((a1 + a2)s + v1v2)
v1^2v2^2 +2a2sv1^2 + 2a1sv2^2 + 4a1a2s^2 = ((a1 + a2)s + v1v2)((a1 + a2)s + v1v2)
v1^2v2^2 +2a2sv1^2 + 2a1sv2^2 + 4a1a2s^2 = ((a1 + a2)s(a1 + a2)s + v1v2(a1 + a2)s) + ((a1 + a2)sv1v2 + v1v2v1v2)
2a2v1^2s + 2a1v2^2s + 2a1a2s^2 = a1^2s^2 + a2^2s^2 + 2v1v2a1s + 2v1v2a2s
2a2v1^2 + 2a1v2^2 + 2a1a2s = a1^2s + a2^2s + 2v1v2a1 + 2v1v2a2
2(a2v1 - a1v2)(v1 - v2) = (a1 + a2)^2s
s = 2(a2v1 - a1v2)(v1 - v2)/(a1 + a2)^2

Answer 2

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