Questions: Calculate the ideal banking angle in degrees for a gentle turn of 1.4 km radius on a highway with a 94.7 km/hr speed limit, assuming everyone travels at the speed limit.

Solution Steps

First, we need to convert the speed from kilometers per hour to meters per second. The speed limit is given as \(94.7 \, \text{km/hr}\).

\[ 94.7 \, \text{km/hr} = 94.7 \times \frac{1000 \, \text{m}}{3600 \, \text{s}} = 26.3056 \, \text{m/s} \]

The formula for the ideal banking angle \(\theta\) is given by:

\[ \tan(\theta) = \frac{v^2}{r \cdot g} \]

where:

• \(v\) is the speed in meters per second,
• \(r\) is the radius of the turn in meters,
• \(g\) is the acceleration due to gravity, approximately \(9.81 \, \text{m/s}^2\).

Substitute the known values:

\[ \tan(\theta) = \frac{(26.3056)^2}{1400 \times 9.81} \]

Calculate the value of \(\tan(\theta)\):

\[ \tan(\theta) = \frac{691.9648}{13734} \approx 0.0504 \]

To find the angle \(\theta\), take the arctangent of the result:

\[ \theta = \arctan(0.0504) \]

Convert the angle from radians to degrees:

\[ \theta \approx 2.8867^\circ \]

Final Answer