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As soon as a traffic light turns green, a car speeds up from rest to 50.0 mi/h with a constant acceleration of 8.10 mi/h·s. In the adjoining bike lane, a cyclist speeds up from rest to 20.0 mi/h with a constant acceleration of 13.0 mi/h·s. Each vehicle maintains a constant velocity after reaching its cruising speed.

(a) For how long is the bicycle ahead of the car?

(b) By what maximum distance does the bicycle lead the car?

  • Physics -

    a) Write equations for distance from the light vs time for bike and car. Call them X1(t) and X2(t)

    Set X1 = X2 and solve for t. These are piecewise continous functions, so you cannot solve single equations.

    If t is in seconds and X in feet,
    X1(car) = 5.94 t^2
    Car acceleration lasts for 6.17 seconds

    X2(bike) = 9.533 t^2
    Bike acceleration lasts for 1.538 s.

    After 1.538s, the bike is ahead by X1-X2 = 22.55 - 14.05 = 8.5 feet. The car is traveling at 18.2 ft/s, which is slower than the bike's 29.3 ft/s but the car still accelerating. Let t' be the time measured after 1.538 seconds, when the bike reaches constant velocity.
    X1 = 14.05 + 18.2 t' + 5.94 t'^2
    X2 = 22.55 + 29.3 t'

    You can now set up a quadratic equation for when X1 = X2

    You can also differentiate X2 - X1 and set the derivative equal to zero to get the maximum separation time and distance.

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