Banked Race Track Physics Problem: Max and Min Speeds without Friction

[shawonna23]

On a banked race track, the smallest circular path on which cars can move has a radius r1 = 111 m, while the largest has a radius r2 = 163 m. The height of the outer wall is 18 m.

(a) Find the smallest speed at which cars can move on this track without relying on friction.

(b) Find the largest speed at which cars can move on this track without relying on friction.

I think that I'm supposed to use this equation:
v= Square root of (r)(g)tan theta, but I don't know how to find theta?

 

[vsage]

you can make a triangle out of the info given. Assuming the base of the track is on the x-z plane, you can create a triangle by taking a cross section pointing radially inward from the circle pointing in the positive y direction. Anyway that's not relevant just trying to give you a reference point. The two legs of the triangle would be (163-111) on the bottom and 18 going up. From this you can find an incline (theta) and you're solving for V.

 

[wais]

Yes, you are correct that the equation v = √(r*g*tanθ) can be used to solve this problem. To find θ, we can use the fact that the height of the outer wall (18 m) is equal to the difference in height between the two circular paths (r2 - r1). So, we can set up the following equation:

tanθ = (r2 - r1)/18

(a) To find the smallest speed, we want to find the minimum value of θ. This occurs when the track is completely flat (θ = 0). So, we can plug in θ = 0 into the equation and solve for v:

v = √(r1*g*tan0) = √(r1*g*0) = 0

Therefore, the smallest speed at which cars can move on this track without relying on friction is 0 m/s.

(b) To find the largest speed, we want to find the maximum value of θ. This occurs when the track is at its steepest angle. So, we can plug in the maximum value of θ (90 degrees or π/2 radians) into the equation and solve for v:

v = √(r2*g*tan(π/2)) = √(r2*g*∞) = ∞

Therefore, the largest speed at which cars can move on this track without relying on friction is infinity. This means that there is no limit to the speed of the cars on this track as long as there is no friction. However, in reality, there will always be some amount of friction present, so the actual maximum speed will be less than infinity.