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A driverless car registers that the traffic lights change to amber 40m ahead. The amber light is a 2s warning before turning red. The car is travelling at 17m/s and can accelerate at 4m/s^2 or brake safely at 8m/s^2. What options does the car have?

Matti P.
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Shah21
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  • S=40m, t=2s, u=17m/s – Shah21 Oct 29 '20 at 13:03
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    Important words to pick up from the text: "car (...) can accelerate (...) or brake". So I guess these two are the options. In the case of accelerating, of course want to pass the lights before they turn to red. For braking, we want the braking distance to be less than 40 m. Let's start with the accelerating case. So suppose that you have an initial speed of 17 m/s and a constant acceleration of 4 m/s^2. How long would it take to cover a distance of 40 meters? – Matti P. Oct 29 '20 at 13:09
  • Can you pls explain in detail – Shah21 Oct 29 '20 at 13:17
  • It takes approx 21 s – Shah21 Oct 29 '20 at 13:18

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Assumming acceleration is constant you can work out how far it would travel when accelerating or decelerating.

For acceleration: Using SUVAT to get: $$v=u+at=25$$ then S=2(17m/s + 25m/s)/2=42 (S=t(u+v)/2) As displacement>40 then car will be able to travel the 40m before the light changes to red

For deceleration: Using SUVAT $S=ut+\frac{1}{2}at^2$ => $$0=4t^2 -17t+40$$ As no real solutions cannot decelerate in time.

Matti P.
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Samuel W
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  • Can you pls explain in detail – Shah21 Oct 29 '20 at 13:17
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    For acceleration you need the distance travelled in those 2 seconds to be greater than 40m because otherwise the red light will show first then by using v=u+at you can find the speed after two seconds then using an equation which gives you S you can then compare S to 40m and as S>40 then the car is able to make it before the light changes red – Samuel W Oct 29 '20 at 13:20
  • I understood. Thank you! – Shah21 Oct 29 '20 at 13:20
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    Then for decceleration I used s=ut+0.5at^2 to find a solution for t but by substituting the known values you get 40=17t-4t^2 which rearranges to 0=4t^2 -17t + 40 which using whatever method you like, eg quadratic formula, you get math error/an imaginary answer so the car cannot brake in time – Samuel W Oct 29 '20 at 13:22
  • Thanks @Matti P – Samuel W Oct 31 '20 at 22:16