Posted by Lisa on Wednesday, April 14, 2010 at 1:19am.
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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