A 1500 kg car is rounding a curve with a radius of 204 meters on a level road. The maximum frictional force the road can exert on the tires of the car totals 4439 N. What is the highest speed at which the car can go to round the curve without sliding?
Answer in m/s.
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To find the highest speed at which the car can go without sliding, we need to determine the maximum centripetal force that can be exerted on the car without exceeding the maximum frictional force.
The centripetal force required for a car to negotiate a curve is provided by the frictional force between the car's tires and the road. It can be calculated using the formula:
F = m * v^2 / r
Where:
- F is the centripetal force
- m is the mass of the car (1500 kg in this case)
- v is the velocity of the car
- r is the radius of the curve (204 meters)
In this case, the maximum centripetal force is limited to the maximum frictional force, which is 4439 N. So we can set up the equation:
4439 N = 1500 kg * v^2 / 204 m
To solve for v, let's re-arrange the equation:
v^2 = (4439 N * 204 m) / 1500 kg
v^2 = (904556 N · m^2) / 1500 kg
Now, divide both sides by 1500 kg:
v^2 = 603.037 N · m^2 / kg
Finally, take the square root of both sides to solve for v:
v = √(603.037 N · m^2 / kg)
Calculating this expression gives us the highest speed at which the car can go to round the curve without sliding.
To find the highest speed at which the car can go to round the curve without sliding, we need to determine the maximum centripetal force that can be generated by the frictional force.
The centripetal force required to keep the car moving in a curve is given by the formula:
Fc = (m * v^2) / r
Where:
Fc is the centripetal force
m is the mass of the car (1500 kg)
v is the velocity of the car
r is the radius of the curve (204 meters)
In this case, the maximum frictional force (4439 N) is equal to the centripetal force:
Fc = 4439 N
Substituting the values into the equation:
4439 N = (1500 kg * v^2) / 204 m
Rearranging the equation to solve for v:
v^2 = (4439 N * 204 m) / 1500 kg
v^2 = 605496 Nm / 1500 kg
v^2 ≈ 403.664 N/m
Taking the square root of both sides to solve for v:
v ≈ sqrt(403.664 N/m)
v ≈ 20.09 m/s
So, the highest speed at which the car can go to round the curve without sliding is approximately 20.09 m/s.