Calculate the maximum deceleration of a car that is heading down a 30.0° slope (one that makes
an angle of 30° with the horizontal) under the following road conditions. You may assume that the
weight of the car is evenly distributed on all four tires and that the coefficient of static friction is
involved—that is, the tires are not allowed to slip during the deceleration. Calculate for a car:
(a) On dry concrete. (b) On wet concrete. (c) On ice, assuming that μs = 0.100, the same as for
shoes on ice. (The last answer will be positive; this means that you can't stop the car, you can only
reduce the downhill acceleration.)
To calculate the maximum deceleration of a car on different road conditions, we need to consider the forces acting on the car and use the laws of physics. The maximum deceleration is achieved when the force of friction between the tires and the road is at its maximum.
The formula for the force of friction is given by:
f = μs * N
Where:
f is the force of friction
μs is the coefficient of static friction
N is the normal force
The normal force N can be calculated as the component of the weight of the car perpendicular to the slope. The weight of the car can be calculated using the formula:
W = m * g
Where:
W is the weight of the car
m is the mass of the car
g is the acceleration due to gravity (approximated as 9.8 m/s^2)
Let's calculate the maximum deceleration on each road condition:
(a) On dry concrete:
Assuming the coefficient of static friction on dry concrete is μs = 0.7, we can calculate the maximum deceleration.
1. Calculate the normal force:
N = m * g * cos(theta)
= m * g * cos(30°)
2. Calculate the force of friction:
f = μs * N
3. Calculate the maximum deceleration:
a = f / m
Substitute the values and calculate the result.
(b) On wet concrete:
Assuming the coefficient of static friction on wet concrete is μs = 0.4, follow the same steps as above to calculate the maximum deceleration.
(c) On ice:
Assuming the coefficient of static friction on ice is μs = 0.1, follow the same steps as above to calculate the maximum deceleration. However, in this case, the result will be positive, indicating that the car cannot stop but only reduce the downhill acceleration.
Remember to substitute the appropriate values for the mass of the car and the angle of the slope in all calculations.