To determine the range of speeds at which a car can safely navigate a curve given a coefficient of static friction ([tex]\(\mu_s\)[/tex]) of 0.36, let's consider the physics involved, specifically centripetal force and friction.
### Key variables:
- [tex]\(\mu_s = 0.36\)[/tex] (Coefficient of static friction)
- [tex]\(r = 100 \, \text{m}\)[/tex] (Assumed radius of the curve)
- [tex]\(g = 9.81 \, \text{m/s}^2\)[/tex] (Acceleration due to gravity)
### Maximum Safe Speed:
The maximum speed at which a car can safely make the curve without sliding outwards is determined by the maximum centripetal force that can be provided by the frictional force. The frictional force acts toward the center of the curve, providing the necessary centripetal force.
[tex]\[ f_{\text{friction}} = \mu_s \cdot N = \mu_s \cdot m \cdot g \][/tex]
where [tex]\(m\)[/tex] is the mass of the car (which will cancel out in our final calculations), and [tex]\(N\)[/tex] (the normal force) is [tex]\(m \cdot g\)[/tex]. This frictional force must provide the centripetal force required for circular motion:
[tex]\[ f_{\text{friction}} = \mu_s \cdot m \cdot g = \frac{m \cdot v^2}{r} \][/tex]
Solving for [tex]\(v\)[/tex]:
[tex]\[ \mu_s \cdot g = \frac{v^2}{r} \][/tex]
[tex]\[ v = \sqrt{\mu_s \cdot g \cdot r} \][/tex]
Substitute the given values:
[tex]\[ v = \sqrt{0.36 \cdot 9.81 \cdot 100} \][/tex]
[tex]\[ v \approx \sqrt{353.16} \][/tex]
[tex]\[ v \approx 18.79 \, \text{m/s} \][/tex]
When expressed using two significant figures:
[tex]\[ v \approx 19 \, \text{m/s} \][/tex]
### Minimum Safe Speed:
For the minimum safe speed, the concept is slightly more nuanced because, theoretically, a car could go infinitely slow and still navigate the curve. However, too slow a speed might not be practical for maintaining a consistent centripetal path, and the practical lower speed needs to be such that the frictional force (if needed) can provide enough centripetal force to keep the car on the path.
Practically, the minimum speed can be considered very close to zero:
[tex]\[ v_{\min} \approx 0 \, \text{m/s} \][/tex]
Thus, the speeds at which a car can safely make the curve fall within the following range:
[tex]\[ 0 \, \text{m/s}, 19 \, \text{m/s} \][/tex]
### Conclusion:
Expressing your answers using two significant figures separated by a comma:
[tex]\[ 0.00 \, \text{m/s}, 19 \, \text{m/s} \][/tex]
