Motion that changes from constant a to constant v

[fizzle45]

Homework Statement

As soon as a traffic light turns green, a car speeds up from rest to 50 mi/h with constant acceleration 9.00mi/h*s. In the adjoining bike lane, a cyclist sppeds up from rest to 20 mi/h with constant acceleration 13mi/h*s. Each vechicle maintains constant velocity after reaching its cruising speed. (a) for what time interval is the bicucle ahead of the car? (b) by what maximum distance does the bike lead the car?

Homework Equations

X(final)=X(initial) + Vx(t)
Vx(final)=Vx(initial)+ax(t)
X(final)=x(initial)+V(initial)t+1/2at^2

The Attempt at a Solution

Im not really sure how to do this problem because the objects go from constant acceleration to constant velocity. Here is my guess.
for the bike it is constant a model until Vx is 20mi/h so when t=V(final)/a the bike changes from constant a model to the constant velocity model. So I plugged t into the constant velocity model for the bike getting X(final)=X(initial)+V(V/a). Then since I want to find when car passes bike i set equation 3 equal to the velocity bike model. x(initialcar)+V(initialcar)t+1/2a(car)t^2 = X(initialbike)+V(bike)(V(bike)/a(bike)). Then if i solve for t i should get the time interval for the bike being ahead of the car but I'm not 100%. To find X initial of bike I took equation 3 for t=v(final)/a. Numbers and conversions aren't important to me as learning how to do these types of problems where the object changes from constant A to V or other way around.

 

[LowlyPion]

You basically have three regimes.

Both accelerating.
One accelerating. One at max.
Both at max.

As you have apparently figured you determine the transitions between the regimes.

To know which equations you need to solve for - i.e. which regime from above - determine also when the slower transitions from acceleration to constant max. If it calculates out further ahead of the faster a one at max speed then you know to use the relationships of the second regime. If it is still behind, then the equations of the last regime, using the initial conditions you've found.

 

[nlightner]

Your approach is correct. To find the time interval when the bike is ahead of the car, you need to equate the positions of both objects at that time. So your equation should be:

X(final bike) = X(final car)

Plugging in the values, you get:

V(bike)^2/(2a(bike)) = V(car)^2/(2a(car))

Solving for t, you get t = (V(bike)/a(bike)) * sqrt(a(car)/a(bike))

This gives you the time interval for the bike being ahead of the car. To find the maximum distance, you can plug this value of t into either of the position equations (X(final)=x(initial)+V(initial)t+1/2at^2) and calculate the distance at that time for both objects. Then take the difference between the two distances to find the maximum distance between them.

Hope this helps!