A car moving with a constant speed of 83 km/h enters a circular, flat curve with a radius of curvature of 0.40 km. If the friction between the road and the car’s tires can support a centripetal acceleration of 1.25 m/s2, without slipping, does the car navigate the curve safely, or does it fly off the road?
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To determine if the car can safely navigate the curve without flying off the road, we need to compare the car's centripetal acceleration with the maximum allowed centripetal acceleration.
Given:
Car's speed = 83 km/h = 83,000/3600 m/s ≈ 23.06 m/s
Radius of curvature = 0.40 km = 0.40 × 1000 m = 400 m
Maximum allowed centripetal acceleration (a) = 1.25 m/s²
The formula for centripetal acceleration is:
a = v² / r
where:
a = centripetal acceleration
v = velocity
r = radius of curvature
Substituting the given values:
a = (23.06 m/s)² / 400 m
a ≈ 1.3353 m/s²
Comparing the calculated centripetal acceleration (1.3353 m/s²) with the maximum allowed centripetal acceleration (1.25 m/s²), we see that the calculated value is greater.
Hence, the car's centripetal acceleration of 1.3353 m/s² exceeds the maximum allowed centripetal acceleration of 1.25 m/s². Therefore, the car cannot navigate the curve safely and is likely to fly off the road.
To determine whether the car can safely navigate the curve or not, we need to calculate the centripetal acceleration and compare it to the maximum centripetal acceleration that the friction can support.
The centripetal acceleration of an object moving in a circular path is given by the equation:
a = (v^2) / r
Where:
a is the centripetal acceleration,
v is the velocity of the car, and
r is the radius of curvature of the curve.
Given:
Velocity of the car, v = 83 km/h
Radius of curvature of the curve, r = 0.40 km
First, we need to convert the velocity from km/h to m/s:
1 km/h = 0.2778 m/s
So, the velocity of the car, v = 83 km/h * 0.2778 m/s/km = 23.05 m/s
Now, let's calculate the centripetal acceleration:
a = (23.05^2) / 0.40
a = 531.46025 / 0.40
a ≈ 1328.65 m/s^2
Next, we compare the calculated centripetal acceleration with the maximum centripetal acceleration that friction can support, given as 1.25 m/s^2. If the calculated centripetal acceleration is less than or equal to the maximum centripetal acceleration, then the car can navigate the curve safely without slipping.
In this case, the calculated centripetal acceleration (1328.65 m/s^2) is greater than the maximum centripetal acceleration (1.25 m/s^2). So, the car cannot navigate the curve safely and will likely slip or fly off the road.
Therefore, the car does not navigate the curve safely and is at risk of flying off the road. Additional caution should be taken by the driver to ensure safety while navigating the curve.