The current fastest roadworthy car known to man is the SSC Tuatara -shown below (picture courtesy of Top Gear: https://www.topgear.com/car-news/supercars/ssc-tuatara-hits-2829mph-second-top-speed-run)- with a whopping top speed of 508 km/hr. It’s easy to believe that achieving this speed was a surprise to the engineers that developed it, but would you believe me if I told you that they could actually accurately estimate the cars top speed before they even built a prototype?

In this article we’ll discuss how this is done and hopefully once you’re done reading you can apply the method effortlessly.

To start, we’ll run through some basic physics. Remember that guy who got hit by an apple…Isaac Newton? Well, he dedicated a lot of his life to the study of physics and it’s principles and because of him humanity made great strides towards understanding how our universe functions. Therefore, 3 basic principles that govern the relationship between an object and the forces that act upon it were named after him.

Newton’s first law states that an object in motion will remain in motion unless acted upon by an equal and opposite force. To elaborate, a car will continue to move down a hill unless you engage its brakes to force it to stop. Newton’s second law states that the net force acting on an object is equal to the product of its mass and acceleration (F=ma). Newton’s third law states that when two bodies interact, they apply forces to one another that are equal in magnitude and opposite in direction. For example if you were to punch a wall you may chip the paint, but you would also feel a lot of pain in your hand because although you imparted a force on the wall the wall imparted an equal force on your hand.

All 3 of these laws are important in the study of physics but to estimate a vehicles top speed we’ll need Newton’s second law. The net force acting on the car is the force supplied by the engine minus the demand forces. The demand forces that affect a cars performance are the vehicle’s weight and its aerodynamic properties. Newton’s second law is displayed in a bit more detail below:

                                       Fs – Fw – Fd = m x a

Where:

Fs= The supplied force measured in Newtons (N)

Fw= The resistance to movement caused by the weight of the car measured in Newtons (N).

Fd= The drag force experienced by the car measured in Newtons (N).

m= The total mass of the car measured in kg.

a= The current acceleration of the car measured in m/s².

It’s important to note that with the above equation we assume that the car is on a level road and is not pulling any extra weight (such as a trailer) because if we didn’t assume this we would have to incorporate it in the equation. Now we’ll dissect each part of the equation starting with the supplied force. This is the maximum force supplied by the engine but usually an engine’s power output is presented in horsepower or kilowatts. First you have to ensure that the power output of the engine is converted to watts, that can be done using an online calculator if you are unsure how to do it. Thereafter, substitute the power value into the equation below .

                                                  Fs = P/V

Where:

P= Maximum power produced (W)

V= Velocity (m/s)

Next we’ll look at the first aspect of a car that hinders motion, its weight (Fw) (also known as rolling resistance). This force is calculated using the equation below. The 0.01 is known as the rolling resistance coefficient for a car driving on asphalt but this value would change if the surface that the car is driving on changed. The other variables are the total mass of the vehicle and the acceleration due to gravity on earth which is approximately 9.81 m/s².

                                     Fw = (0.01) x m x g 

Where:

m= Weight of the car in kg.

g= Acceleration due to gravity.

Drag tends to be the biggest impedance to motion at high speeds. As seen in the below equation the velocity of the car is squared which means it has the biggest affect on the equation. The density of air is usually assumed to be 1.225 kg/m³ at sea level but the density of air decreases with elevation so that may be something to keep in mind if you want to be really accurate with the calculation. The coefficient of drag is a constant that is only derived via experimental analysis so it will differ depending on the car but a value of 0.3 can be assumed if you don’t know the exact value. The frontal area is exactly what it sounds like, it’s the area of the vehicle that is considered the front of the vehicle. The picture below should give you an idea of what the frontal area is (picture courtesy of: www.buildyourownracecar.com).

                               Fd = (0.5) x p x Cd x A x V²                                    

Where:     

p= Density of air measured in kg/m³.

Cd= Coefficient of drag.

A= Frontal area of the vehicle measured in m².

V= Velocity measured in m/s².

Lastly, we’ll discuss the m x a part of the equation. Since we’re trying to calculate top speed and as the concept clearly implies that this is the cars maximum speed we can deduce that the car would not be able to accelerate anymore so the acceleration would be equal to 0. After all that we have expanded on, the equation now looks like this:

(P/V) – [(0.01) x m x (9.81)] – [(0.5) x (1.225) x (0.3) x A x V²] = 0

Since the equation has been simplified, all that we have to do is substitute the car’s power output, weight, frontal area and solve for the velocity (V). The answer you get will be the theoretical top speed of the car. Now that we’ve seen the physics behind vehicle engineering, you can understand why sports cars have such aerodynamic shapes instead of insanely large engines.