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

In summary, on a banked race track with a radius of 111 m and 163 m and a height of 18 m, the smallest and largest speeds at which cars can move without relying on friction can be found using the equation v= √(r)(g)tanθ, where θ is the incline of the track. To find θ, you can create a triangle using the given information and solve for V.
  • #1
shawonna23
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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?
 
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  • #2
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.
 
  • #3


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.
 

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

1. What is a "banked curve" in physics?

A banked curve is a curved section of a road or track that is intentionally sloped or tilted to one side. This design allows vehicles or objects to travel around the curve with less friction and a more stable trajectory.

2. How is the angle of banking determined in a banked curve?

The angle of banking in a curve is determined by the speed at which the vehicle or object will be traveling, the radius of the curve, and the coefficient of friction between the tires or wheels and the surface of the curve. This angle is typically calculated using the formula tanθ = (v^2) / (rg), where θ is the angle of banking, v is the speed, r is the radius of the curve, and g is the gravitational acceleration.

3. What factors affect the maximum speed at which a vehicle can safely travel around a banked curve?

The maximum speed at which a vehicle can safely travel around a banked curve is affected by the angle of banking, the coefficient of friction between the tires or wheels and the surface of the curve, and the mass and size of the vehicle. Other factors such as the condition of the tires and the driver's skill also play a role.

4. How does a banked curve affect the centripetal force on a vehicle?

A banked curve allows for the centripetal force on a vehicle to be directed at an angle towards the center of the curve, rather than purely perpendicular to the surface. This reduces the amount of friction and allows for a higher speed around the curve.

5. What is the difference between a "superelevation" and a "banked curve"?

Superelevation is the process of raising the outer edge of a road or track to create the angled surface of a banked curve. A banked curve is the overall design of a curved section of road or track that allows for higher speeds and reduced friction. So, superelevation is a step in creating a banked curve.

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