A roadway for stunt drivers is designed for race cars moving at a speed of 95 m/s. A curved
section of the roadway is a circular arc of 760 m radius. The roadway is banked so that a vehicle
can go around the curve with the friction force from the road equal to zero. At what angle is the
roadway banked?
v=95m/s
r=760 m
mg tan theta = mv^2/r
theta =?
theta = arctan(v^2/gr)
theta = arctan(95^2/(9.8*760))
theta = 8.7 degrees
To find the angle at which the roadway should be banked, we can use the equation:
mg tan(theta) = mv^2 / r
Here's how to solve the problem step by step:
Step 1: Identify the given values:
- Velocity (v) = 95 m/s
- Radius (r) = 760 m
- Acceleration due to gravity (g) = 9.8 m/s^2
Step 2: Substitute the given values into the equation:
mg tan(theta) = mv^2 / r
Note: The mass of the vehicle (m) cancels out on both sides of the equation, so we don't need to know it.
Step 3: Rearrange the equation to solve for the angle (theta):
tan(theta) = (v^2 / r) * (1 / g)
Step 4: Calculate the value inside the parentheses:
(v^2 / r) * (1 / g) = (95^2 / 760) * (1 / 9.8) ≈ 9.48
Step 5: Take the inverse tangent (arctan) of the value calculated in step 4 to find the angle (theta).
theta ≈ arctan(9.48) ≈ 84.7 degrees
Therefore, the roadway should be banked at an angle of approximately 84.7 degrees.
To find the angle at which the roadway is banked (θ), we can use the formula:
m * g * tan(θ) = m * v^2 / r
Where:
m is the mass of the race car,
g is the acceleration due to gravity (approximately 9.8 m/s^2),
v is the velocity of the race car (95 m/s), and
r is the radius of the curved section of the roadway (760 m).
Since the mass of the race car cancels out on both sides of the equation, we can simplify it to:
g * tan(θ) = v^2 / r
Now, we need to rearrange the equation to solve for θ.
First, divide both sides of the equation by g:
tan(θ) = (v^2 / r) / g
Then, take the inverse tangent (tan^-1) of both sides to isolate θ:
θ = tan^-1[(v^2 / r) / g]
Now, substitute the given values into the equation to find θ:
θ = tan^-1[(95^2 / 760) / 9.8]
Calculating using a calculator:
θ ≈ tan^-1[0.149]
θ ≈ 8.56 degrees
Therefore, the roadway needs to be banked at an angle of approximately 8.56 degrees.