A motorcyclist is traveling at 50.7 mph on a flat stretch of highway during a sudden rainstorm. The rain has reduced the coefficient of static friction between the motorcycle's tires and the road to 0.072 when the motorcyclist slams on the brakes.
(a) What is the minimum distance required to bring the motorcycle to a complete stop? m
(b) What would be the stopping distance if it were not raining and the coefficient of static friction were 0.580? m
To solve this problem, we can use the equations of motion and the concept of friction to find the minimum stopping distance for the motorcycle.
First, let's calculate the minimum stopping distance when it's raining and the coefficient of static friction is 0.072.
(a) Minimum stopping distance during rain (friction coefficient = 0.072):
1. The first step is to convert the velocity to meters per second (m/s):
Speed in m/s = Speed in mph × 0.44704
Speed in m/s = 50.7 mph × 0.44704 = 22.7 m/s
2. The equation for the stopping distance is given by:
Stopping distance = (Initial velocity^2) / (2 * acceleration)
3. The acceleration can be calculated using the equation:
friction force = coefficient of friction * normal force
In this case, the normal force is equal to the weight of the motorcycle:
Normal force = mass of motorcycle * gravity
acceleration = friction force / mass of motorcycle
4. The friction force can be determined using the equation:
friction force = (normal force) * (coefficient of friction)
Substituting in the values, we have:
acceleration = (mass of motorcycle * gravity * coefficient of friction) / mass of motorcycle
acceleration = gravity * coefficient of friction
Note: The mass of the motorcycle cancels out in the equation.
5. The acceleration is equal to the negative of the deceleration for this case:
acceleration = -deceleration
6. Substituting the values into the equation for the stopping distance, we have:
Stopping distance = (velocity^2) / (2 * acceleration)
Stopping distance = (22.7 m/s)^2 / (2 * (-gravity * coefficient of friction))
Stopping distance = (515.29 m^2/s^2) / (2 * (-9.8 m/s^2 * 0.072))
Stopping distance = (515.29 m^2/s^2) / (-1.4112 m/s^2)
Stopping distance ≈ -365.27 m^2
Since the distance cannot be negative, we take the absolute value:
Minimum stopping distance during rain ≈ 365.27 m
So, the minimum distance required to bring the motorcycle to a complete stop during the rain is approximately 365.27 m.
(b) Now, let's calculate the stopping distance when it's not raining, and the coefficient of static friction is 0.580.
1. We can use the same formula for stopping distance:
Stopping distance = (velocity^2) / (2 * acceleration)
2. Using the given values, we find:
Stopping distance = (22.7 m/s)^2 / (2 * (-gravity * coefficient of friction))
Stopping distance = (22.7 m/s)^2 / (2 * (-9.8 m/s^2 * 0.580))
Stopping distance = (515.29 m^2/s^2) / (2 * (-5.684 m/s^2))
Stopping distance ≈ 45.36 m
Therefore, the stopping distance if it were not raining and the coefficient of friction were 0.580 is approximately 45.36 m.