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I have an object that can accelerate at a known rate and decelerate at a different known rate that needs to cover a known distance as quickly as possible.

It must start and end stationary so it would have to accelerate at its acceleration rate for a certain (unknown) distance and then immediately start decelerating at its deceleration rate for the rest of the journey.

How can I work out the time the whole journey would take?

Omegatron
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1 Answers1

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Let $t_1$ be the time your object accelerates. Write down a function $f(t_1)$ that gives the time it takes to decelerate to a standstill afterwards. Write down a function $h(t_1)$ that gives the distance covered by this. Set $h(t_1)$ equal to your desired distance, and solve for $t_1$. Plug the solution you find into $t_1+f(t_1)$ and evaluate.

hmakholm left over Monica
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  • "Set h(t1) equal to your desired distance, and solve for t1"

    Wouldn't that give a time as if it had been decelerating for the whole journey?

    Following this I've got t1=square root(s/(-0.5a1^2)) and f(t1)=-a1t1/a2 which doesn't seem to work but it's possible I worked it out wrong.

    (I have no idea how to do fancy formatting or line-breaks in comments, sorry).

     – Omegatron Jan 04 '16 at 20:10
  • @Omegatron: $h(t_1)$ is supposed to be the distance covered by both acceleration and deceleration. – hmakholm left over Monica Jan 04 '16 at 21:13
  • Ah thanks, got it working now. – Omegatron Jan 05 '16 at 00:30