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I came across a fairly basic kinematics question in the first chapter of "An Introduction to Mechanics, 2nd Edition" by Kleppner & Kolenkow:

"A sportscar, Electro-Fiasco I, can accelerate uniformly to 100 km/h in 3.5 s. Its maximum braking rate cannot exceed 0.7g. What is the minimum time required to go 1.0 km, assuming it begins and ends at rest?"

In the solution manual, the author assumes that in order to minimize trip duration, the car must accelerate maximally in the positive direction until some time t, at which point acceleration immediately changes sign and the car brakes maximally until it stops. In other words, the author assumes that the car does not coast between speeding up and slowing down. Please prove, as rigorously as possible, that this assumption is viable :)

When I sketch a trapezoidal function of velocity vs. time, it's easy to see that the area of the trapezoid is the distance traveled. Why must the upper width be zero? This makes intuitive sense to me, but I otherwise can't justify my assumption.

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  1. Draw the trapezoid for some arbitrary constant-speed interval between the speeding and slowing phases.
  2. Continue the tilted lines of constant acceleration until they meet (somewhere above the trapezoid's upper base).
  3. We get a shape which has the same time duration (lower base), but a greater area.
  4. We conclude that for a shape of maximal area at any given time duration, we should draw a triangle, and not a trapezoid.
  5. Therefore the solution of minimal time for a given area (distance) cannot be a trapeziod, and must be a triangle
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