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Math Help - Antiderivative and acceleration question

  1. #1
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    Antiderivative and acceleration question

    At the instant the traffic light turns green, an automobile that has been waiting at an intersection starts moving forward with a constant acceleration of 6 feet per second squared. At the same instant, a truck traveling with a constant velocity of 30 feet per second overtakes and passes the car. How far beyond its starting point will the automobile overtake the truck?

    I'm so lost, any help would be appreciated!

    EDIT: Sorry this is my first time posting, and I did not realise I should show work. I also do not know how to write in math text. All I know is that the integral of acceleration in equal to velocity, and the integral of velocity is equal to position. I still do not know how to approach this problem, however.
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  2. #2
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    The equation you need here is the can be derived in two lines from the fact that the acceleration is the double derivative of the displacement. And for both vehicles the acceleration is constant (either 6 or 0 ft/s).

    \frac{d^2s}{dt^2} = a.

    Integrating

    \frac{ds}{dt} = v_0 + at,

    where v_0 is the velocity of the vehicle at the time t=0. Integrating again gives

    s = s_0 + v_0 t + \frac{1}{2} at^2,

    where s_0 is the displacement of the vehicle at the time t=0. In this problem they are both at the traffic light which we will take as s=0 and so s_0 can be ignored in this solution.

    So the displacement of the car s_c is easy as in this case v_0 = 0 and a = 6, so

    s_c = 3t^2.

    And in the case of the truck s_t, v_0 = 30 and a = 0

    s_t = 30t.

    Now we want the time t_1 where the car overtakes the truck. IE displacements are equal therefor s_c = s_t

    3t_1^2 = 30t_1

    t_1 = 10

    And finally to find what that distance is s_1 you can substitute t=t_1 into either the s_c or the s_t equation as they will give the same answer.

    s_1 = 30 t_1 = 300.

    So the car will overtake the truck 300 ft from the traffic light.

    To make sure you have understood this, try working out how fast the car will be travelling at the point that it overtakes the truck.

    Hope this helps.
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