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?
We need the horizontal acceleration so,
ax=Fx/m
Since Fx, or the horizontal forces acting on the car is composed of the air resistance and the force of static friction,
ax=(fs-1300N)/m
Looking for fs(static friction):
fs=us*N
N=mg+3150N
=(552kg)(9.8)+3150N
=8559.6N
fs=(0.967)(8559.6N)
=8277.13N
Solving for ax:
ax=(8277.13-1300N)/553
ax=12.54m/s2
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.
The forces acting on the car are:
1. Weight (mg) = 552 kg × 9.8 m/s^2 = 5409.6 N (downwards)
2. Normal Force (N) = Weight (mg) + Downforce = 5409.6 N + 3150 N = 8559.6 N (upwards)
3. Horizontal Force (F) = Horizontal Air Resistance = 1300 N (opposite to the direction of motion)
Since the car is accelerating without its tires slipping, the maximum horizontal frictional force (f) between the tires and the track equals the maximum static friction (fs) multiplied by the normal force (N).
Therefore, fs = coefficient of static friction (μ) × N = 0.967 × 8559.6 N = 8286.04 N.
The maximum acceleration (a) is given by the equation: f = ma.
Rearranging the equation, a = f/m = fs/m = 8286.04 N / 552 kg = 15.00 m/s^2 (approximately).
Therefore, the magnitude of the maximum acceleration the car can achieve without its tires slipping is 15.00 m/s^2.
To find the maximum acceleration at which the car can speed up without its tires slipping, we need to consider two forces: the downforce and the horizontal air resistance force.
First, let's calculate the maximum friction force that the tires can exert on the car. We can do this by multiplying the coefficient of static friction (μ) by the normal force, which is the sum of the car's weight and the downward force.
Normal force = weight + downforce
Normal force = mass * acceleration due to gravity + downforce
Normal force = 552 kg * 9.8 m/s^2 + 3150 N
Next, let's calculate the maximum friction force:
Maximum friction force = coefficient of static friction * normal force
Maximum friction force = 0.967 * (552 kg * 9.8 m/s^2 + 3150 N)
Now, let's consider the horizontal air resistance force. This force opposes the car's motion and can decrease its acceleration. We need to subtract the horizontal air resistance force from the maximum friction force to determine the net force available for acceleration.
Net force available for acceleration = maximum friction force - horizontal air resistance force
Net force available for acceleration = (0.967 * (552 kg * 9.8 m/s^2 + 3150 N)) - 1300 N
Finally, we can determine the maximum acceleration using Newton's second law:
Maximum acceleration = Net force available for acceleration / mass
Maximum acceleration = (0.967 * (552 kg * 9.8 m/s^2 + 3150 N) - 1300 N) / 552 kg
Solving this equation will give us the magnitude of the maximum acceleration at which the car can speed up without its tires slipping.