Friction in the Acceleration of Tesla Model S

Tesla Model S (Credit: Tesla.com)

Elon Musk revolutionized the way people perceived electric vehicles with his company Tesla Motors.  Musk created Tesla with the vision of creating powerful, visually appealing, and functional electric cars.  The Tesla Model S was the first Tesla vehicle to truly embody this vision.  Over time, the car has become increasingly more powerful.  The highest end Model S can now do 0-60 mph in 2.5 seconds.  Using the material recently taught in class, the force of friction and the friction coefficient can be estimated for the Tesla Model S.

Kinematics of the Tesla Model S
Assuming constant acceleration, the acceleration can be found from kinematics.

Tesla Model S and Lamborghini Huracan LP610-4 drag-racing
(Credit: Road & Track)

• ∆X = 
• Vi = 0
• Vf = 60mph = 96.56 km/h = 26.82 m/s
• a = ?
• t = 2.5 seconds

Vf = Vi + at

26.82 = 0 + 2.5a

26.82 = 2.5a

a = 10.728 m/s^2

Free-Body Diagram and Sum of Forces

Next, a free-body diagram can be drawn to represent the forces acting upon the car.  There is Fg (the force of gravity), N (the normal force), and f (friction).  It should also be noted that the mass of the car is 4,647.3 lbs, which is 2108 kg.

∑Fy = N - Fg = 0
∑Fy = N = mg
N = (2108)(9.8) = 20658 N

∑Fx = f = ma
∑Fx = μs*N ≤ ma
μs(20658) ≤ (2108)(10.728)
(20658)μs ≤ 22615
μs ≤ 1.09
f = 22615 N

Accuracy of Results

Graph of the acceleration vs. time of a
Model S and a Hellcat (Credit: Consumer Reports)

The values obtained above are inaccurate due to a few factors.  First, cars rarely accelerate constantly.  Since acceleration would not be constant, kinematics could not be used.  Second, there are other forces acting on the car, including drag.  Since these forces are not factored in, the answers obtained above are too high.  In fact, the coefficient of friction is impossible since the coefficient must be between 0 and 1.





Sources
http://www.animations.physics.unsw.edu.au/jw/car-physics.htm
https://www.tesla.com/support/model-s-specifications
https://www.forbes.com/sites/chadorzel/2016/08/25/can-a-tesla-model-s-really-accelerate-faster-than-gravity/#28adac8520a2