Time of cars lapping on second/third lap

Sep 2018
Hi guys,
I've got a problem involving a track and two cars. The problem appears to be a fairly simple simultaneous equation but actually turns out to be a bit more complex.

Two toy cars start from the same position and race around a track. Both cars accelerate for 3 seconds. Car A accelerates at a rate of 2m/s/s while car B accelerates at a rate of 3.2m/s/s. The radius of the track is 2.5m. At what time does car b overtake car A.

The problem in encountering in the maths is that car B completes a lap before overtaking and therefor its displacement resets to 0 while car A continues about its journey.

Anyone have any ideas?


PHF Helper
Jun 2010
Morristown, NJ USA
The trck is to realize that the total distance travelled by car B must be $2 \pi R = 15.71$ meters further than the distance traveled by car A. B’s position at t=3 seconds is 14.4 m, and A’s is 9 m. So B has is 5.4 m ahead of A, and must make up an additional 15.71 - 5.4 = 10.31 meters in order to lap A, while traveling at a speed that is 3.6 m/s faster. Can you take it from here?
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