Forces & Acceleration
Newton’s 2nd Law of motion, net force = mass x acceleration , states that any object with a non-zero net force acting on it will accelerate [1] and that with equal forces acting on them, a lighter object will accelerate at a greater rate than a more massive one (highlighting the importance of the use of light materials in F1).
Both force and acceleration are vector quantities (they contain information about magnitude and direction of the quantity), therefore, Newton’s 2nd law tells us that the object will accelerate in the direction of the net force.
Calculating the net force from vector quantities.
This is a lot simpler than it may sound, we just need to pay attention to the direction in which the forces act. First we resolve vertically, we attribute a positive value to forces acting upwards and a negative value to those acting downwards, then do the same for the horizontal.
Driving & Braking Forces
The engine produces a torque on the wheels (which is equal to the radius of the wheels multiplied by the tangential force) the only tangential force acting on the wheels is the frictional force with the track. Applying Newton’s third law we see that as the car exerts a force pushing the track backwards, the track must exert an equal magnitude force forward on the car and therefore the car accelerates forwards. The maximum force which the car can exert on the track, before the wheels slip and spin, is the maximum frictional force (as given above). Therefore, increasing the downforce increases the maximum frictional force and in turn the acceleration (this assumes that the maximum driving force from the engine is not the limiting factor which is true for relatively low speeds).
If the car is braking the same principle is applied in reverse, the tyres exert a force pushing the track forward and in turn the track pushes the car backwards, creating a net force backwards and causes the car to decelerate. Again, the maximum braking force which can be applied is equal to the maximum static friction. If this is exceeded, the car will skid.
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Cornering
Maximum Cornering Speed
To attain the maximum cornering speed of the car we must first substitute in the equation for reaction force R as this has a velocity dependence and then re-arrange to get an expression for the maximum speed at which the car can corner.
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Maximum cornering speed in the model
As the maximum speed is a real observable we require that the equation for maximum cornering speed gives us a real number, that means that the denominator in the equation must be positive which although for all practical scenarios this is satisfied, theoretically it is possible. Where does this model fall down then?
It is due to mass not remaining constant at high speeds. As the denominator approaches zero, this (classically) increases the speed towards infinity. However, as stated by Einstein’s theory of special relativity, as the speed approaches the speed of light, the mass will increase without limit meaning that the velocity will reach a finite, real value.
The model does not include the effects of special relativity since they are not relevant at the speeds that an F1 car achieves. The assumption made by the model to avoid throwing up an error if the denominator becomes negative is to return the speed of light, the precise value of this number is not relevant since the car is still limited by its maximum speed.