Answer :
To find the maximum speed at which a car can navigate a curve without skidding, we need to use the concept of centripetal acceleration. Here are the steps to solve this:
Understanding Centripetal Acceleration:
Centripetal acceleration (
a_c) is given by the formula:[tex]a_c = \frac{v^2}{r}[/tex]
where:
- [tex]v[/tex] is the speed of the vehicle,
- [tex]r[/tex] is the radius of curvature of the curve.
Maximum Centripetal Acceleration:
The maximum centripetal acceleration can be determined by the grip of the tires with the pavement. In this case, the maximum centripetal acceleration is given as 0.5 g, where g is the acceleration due to gravity (approximately 9.81 m/s²).Therefore,
[tex]a_{max} = 0.5 \cdot g = 0.5 \cdot 9.81 \approx 4.905 \text{ m/s}^2[/tex]
Setting Up the Equation:
Now we can set this equal to the centripetal acceleration formula:[tex]\frac{v^2}{r} = a_{max}[/tex]
Substituting the value of [tex]a_{max}[/tex] and the given radius (97.6 m) into the equation, we have:
[tex]\frac{v^2}{97.6} = 4.905[/tex]
Solving for Speed:
Rearranging the equation to solve for speed (v):[tex]v^2 = 4.905 \cdot 97.6[/tex]
[tex]v^2 \approx 479.388[/tex]
Taking the square root:
[tex]v \approx \sqrt{479.388} \approx 21.9 \text{ m/s}[/tex]
Converting from m/s to km/h:
To convert the speed from meters per second to kilometers per hour, use the conversion factor (1 m/s = 3.6 km/h):[tex]v \approx 21.9 \text{ m/s} \cdot 3.6 = 78.84 \text{ km/h}[/tex]
Final Answer:
The maximum speed at which the curve can be taken without skidding is approximately 78.84 km/h.