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28 Mar 2018

Solutions – Fundamental Applied Maths 2nd Ed

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All Fundamental Applied Maths Solutions are subject to:

© Oliver Murphy 2011, shared with the permission of Folens publishers 2018
That is, I have been given permission from Folens to share these solutions with my students only.
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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 1 and car 2 (1 is behind 2) is s_1 and s_2 respectively then their separated distance is

          \[s_2 - s_1\]

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

          \[\frac{ds_2}{dt}-\frac{ds_1}{dt}=0\]

      which means

          \[v_2 - v_1 = 0\]

      which then of course means

          \[v_1=v_2 .\]

      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

          \[\frac{d^2s_2}{dt^2}-\frac{d^2s_1}{dt^2}<0\]

      that is

          \[a_2 < a_1 .\]

      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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