Question

You are stopped at the end of the merge lane at the entrance to a motorway because traffic is dense with cars at a distance of 200m from each other with no larger gaps in sight. They are all travelling at a steady 87 km/h. You have no choice but to join traffic and accelerate from rest as fast as your 15-year old Volkswagen allows (0-114 km/h in 12s). You start accelerating directly after a car passes you. Assuming that the following driver does not slow down to let you in, how close does he come up behind you?

Answer 1

87 km/h = 87/3.6 = 24.2 m/s
114 m/h/3.6 = 31.7 so 0 to 31.7 m/s in 12 s = 2.64 m/s^2
now
in time t
your speed u = 2.64 t
your distance = 1.32 t^2
how long do you take to get to 87 km/h = 24.2 m/s?
2.64 t = 24.2
t = 9.17 seconds to get to speed
you went
1.32 (9.17)^2 = 111 meters (about a football field)

how far did he go in that 9.17 s?
24.2 * 9.17 = 222 meters
so he is 22 meters behind you when you are both at the same speed

Answer 2

My last line is wrong
he is at 222 - 200 = 22 meters from the intersection
you are at 111 meters from the intersection

Answer 3

So why is it Distance = 1.32t^2 ???