A race car track has been constructed with the a bank angle of 33.6° so there is no friction force. The race car track has a curve radius of 390. m. What speeds, for the race cars, has the track been designed to accommodate?
Question
A race car track has been constructed with the a bank angle of 33.6° so there is no friction force. The race car track has a curve radius of 390. m. What speeds, for the race cars, has the track been designed to accommodate?
Solution
To solve this problem, we can use the physics concept of banking of roads, which is used to prevent friction between the tires and the road. The formula for the speed of a car on a banked road without friction is:
v = sqrt(r * g * tan(θ))
where:
- v is the speed of the car,
- r is the radius of the curve,
- g is the acceleration due to gravity (approximately 9.8 m/s²), and
- θ is the bank angle.
Given:
- r = 390 m,
- g = 9.8 m/s², and
- θ = 33.6°.
We first need to convert the angle from degrees to radians because the trigonometric functions in most calculators use radians. We can do this by multiplying by π/180:
θ = 33.6° * π/180 = 0.586 radians.
Now we can substitute the values into the formula:
v = sqrt(390 m * 9.8 m/s² * tan(0.586 radians)).
Calculating this gives:
v = sqrt(390 m * 9.8 m/s² * 0.631) = sqrt(2413.714 m²/s²) = 49.13 m/s.
So, the track has been designed to accommodate race cars traveling at a speed of approximately 49.13 m/s.
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