To determine the speed at which a car can travel around curve B without relying on friction, we need to apply the concept of centripetal force and the equation for frictionless banking of curves.
The equation for frictionless banking is given by:
tan(θ) = v^2 / (g * r)
Where:
- θ is the angle of the banked curve,
- v is the velocity of the car,
- g is the acceleration due to gravity (approximately 9.8 m/s^2),
- r is the radius of the banked curve.
We are given that curve A is banked at 13 degrees and the car can travel around it at a speed of 18 m/s. We need to find the speed the car can travel around curve B, which is banked at 19 degrees.
Let's assume the radius of both curves is the same.
For curve A:
θA = 13 degrees
vA = 18 m/s
Using the equation for curve A, we can rearrange it to solve for rA:
rA = vA^2 / (g * tan(θA))
Next, we can use the same equation for curve B, where θB = 19 degrees, to solve for vB:
vB = sqrt(rB * g * tan(θB))
To find vB, we need to determine the radius of curve B. Since both curves have the same radius, we can equate the expressions for the radii of curve A and curve B:
rA = rB
Substituting the expression for rA into the equation for curve B, we have:
sqrt(rA * g * tan(θB)) = vB
We can now substitute the known values into the equation and solve for vB.
1. Calculate rA:
rA = (18 m/s)^2 / (9.8 m/s^2 * tan(13 degrees))
2. Calculate rB:
rB = rA
3. Calculate vB:
vB = sqrt(rB * 9.8 m/s^2 * tan(19 degrees))
By following these calculations, we can determine the speed at which the car can travel around curve B without relying on friction.