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All things Mathematical
28 Mar 2018

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2 Responses

1. Sam

Hi. How is he greatest gap between two particles when their speeds are equal. If one is accelerating and the other decelerating then wouldn’t there be a bigger gap?

1. stephen

One argument comes from calculus, if the position of car and car ( is behind ) is and respectively then their separated distance is

and according to calculus their minimum or maximum distance apart will be when

which means

which then of course means

It is not clear from this argument alone which this will be, minimum or maximum. That depends on the acceleration but it will either be a minimum a maximum. For a maximum, which is what you want, the second derivative must be negative, that is

that is

So the first car must be have a greater acceleration than the second car at the moment their velocities are equal.

You are right, technically, equal velocities alone is not enough for a maximum.

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