Air rushing over the wings of high-performance race cars generates unwanted horizontal air resistance but also causes a vertical downforce, which helps cars hug the track more securely. The coefficient of static friction between the track and the tires of a 552-kg car is 0.967. What is the magnitude of the maximum acceleration at which the car can speed up without its tires slipping when a 3150-N downforce and an 1300-N horizontal air resistance force act on it?
To find the magnitude of the maximum acceleration at which the car can speed up without its tires slipping, we need to consider the forces acting on the car and compare the maximum static frictional force with the net force.
Given:
Mass of the car (m) = 552 kg
Coefficient of static friction (μ) = 0.967
Downforce (Fd) = 3150 N
Horizontal air resistance force (Fh) = 1300 N
The maximum static frictional force (Fsf) can be calculated using the coefficient of static friction (μ) and the normal force (Fn).
The normal force (Fn) is the gravitational force acting on the car, and it can be calculated as the sum of the weight (mg) and the vertical downforce (Fd).
Fn = mg + Fd
Fn = (552 kg)(9.8 m/s^2) + 3150 N
Next, we need to find the maximum static frictional force (Fsf) by multiplying the coefficient of static friction (μ) by the normal force (Fn).
Fsf = μ * Fn
Fsf = 0.967 * (mg + Fd)
Now, let's calculate the net force (Fnet) acting on the car by subtracting the horizontal air resistance force (Fh) from the propulsive force.
Fnet = Fd - Fh
Fnet = 3150 N - 1300 N
Finally, we can determine the maximum acceleration (a) by dividing the net force (Fnet) by the mass of the car (m).
a = Fnet / m
a = (3150 N - 1300 N) / 552 kg
So, the magnitude of the maximum acceleration at which the car can speed up without its tires slipping is the value calculated above.