A race car track has been constructed with the a bank angle of 32.7° so there is no friction force. The race car track has a curve radius of 394 m. What speeds, for the race cars, has the track been designed to accommodate?
gravity force down slope = m(acceleration down slope)
m g sin 37.5 =(m v^2/r) cos 37.5
g tan 37.5 = v^2/r
v^2 = r g tan 37.5 = 394*9.81 tan 37.5
v = 54.5 m/s
To determine the speeds for the race cars that the track has been designed to accommodate, we need to consider the centripetal force acting on the cars as they navigate the curved track.
The centripetal force is the force that keeps an object moving in a circular path and is given by the equation:
F = (m * v^2) / r
Where:
F is the centripetal force,
m is the mass of the car,
v is the velocity of the car, and
r is the radius of the curve.
Assuming the only force acting on the car is the centripetal force and there is no friction, we can equate it to the weight of the car:
F = m * g
Where:
m is the mass of the car and
g is the acceleration due to gravity.
Since the centripetal force and the weight of the car are equal, we can set the equations equal to each other:
(m * v^2) / r = m * g
We can cancel out the mass (m) from both sides of the equation:
v^2 / r = g
Now, we can solve for the velocity (v):
v^2 = g * r
v = sqrt(g * r)
Substituting the given values:
g = 9.8 m/s^2 (acceleration due to gravity),
r = 394 m (radius of the curve),
v = sqrt(9.8 * 394)
v ≈ 62.34 m/s
Hence, the track has been designed to accommodate speeds of approximately 62.34 meters per second for the race cars.
To find the speed at which a race car can safely travel on a banked curve without any friction, we can use the following formula:
v = √(rgtanθ)
where:
v = velocity of the race car
r = radius of the curve
g = acceleration due to gravity
θ = angle of the track's bank
In this case, the angle of the track's bank is given as 32.7°, and the radius of the curve is given as 394 m.
First, we need to convert the angle from degrees to radians:
θ = (32.7° * π) / 180°
Next, we can substitute the given values into the formula:
v = √(394 * g * tan(θ))
Now, we can calculate the speed:
v = √(394 * 9.8 * tan(32.7° * π / 180°))
Using a value of 9.8 m/s^2 for acceleration due to gravity, and assuming π is approximately 3.14159, we can calculate the speed.