www.vimeo.com/332975728/771e62bd78

Please see the above video on Vimeo which is my final submission for my Scholar’s Project. Below are some additional notes further explaining the methodology and the bibliography, please see also the Excel spreadsheets for the workings.

Methodology
Initially I read the book ‘Physics for Gearheads’. The project is largely derived from information in chapters 10 and 11 which shows how to calculate lap times on a rectangular track with linear acceleration.

From there, I furthered the ideas by creating a more complex track, and extrapolated the necessary formulas, using various other sources where required (see bibliography). Instead of having linear acceleration, I based it on a real car with non-linear lateral acceleration. I selected the 2017 Dodge Viper ACR Extreme because it is a high performance track that has more downforce than most, meaning the difference in lateral acceleration would be higher at faster speeds.

I utilised excel spreadsheets to calculate the values, linking the calculations together (see final 4 spreadsheets).

I utilised the animation and rendering programme Blender to produce the video. I created all the lettering required by individually modelling each letter (as Blender does not have a text function to write formulae). Blender models by utilising vertices to create objects which then are used to compose an animation by assigning them positions, rotations and scales in each frame. There are 8, 765 frames in this animation. I animated all the track, formulae and labels that I had individually modelled to create and then render the individual png pictures, which are then put back into the video editor in Blender. Then I recorded the voice over and laid it over the animation along with the backing track ‘Sky Skating’ by Geographer.

I would like to thank Prof. Colin Garner and his research team at Loughborough for their input and assistance in figuring out how to construct an equation for the change of line in and out of the crossover straights (see formula on next page)

Scholars Project Script – to accompany the video.

Hi I’m Caleb and this is my scholars project on ‘Calculating lap times around a race track’.
In this video I’ll show you all the steps you need to take to be able to calculate how fast a car can go around a track. The car I will be using for this project is the 2017 Dodge Viper ACR Extreme.
Now firstly, you’ll need a set track. I have designed a track here, with 8 corners, a length of 3.222km (about the same as the Monaco F1 track) and the longest straight being 631m. Before you can begin the calculations you will have to know the internal radius, width and angle of each corner and the length of each straight, as well as the width of the car, torque curve, gear ratios, lateral acceleration and downforce graph.
Firstly, we need to find the fastest line to take round the track. This is called the racing line, it’s the widest radius curve you can get within the track limits. The formula for this is shown here, you need to know the inner radius of the corner, the effective track width (which is just the track width take the car width) and the angle of the corner. If we substitute in the values for this corner, we get a racing line radius of 59.14 metres. If we do this for all the other corners, we get these values.
Now we need to know how fast the car can go around these corners using this formula here. But the formula here doesn’t account for lateral acceleration depending on speed due to downforce having a greater effect at higher speeds. For that we are going to need a graph plotting speed over acceleration. The first line on this graph is the lateral acceleration over speed, the lateral acceleration starts increasing at roughly 27 metres per second, or 60 miles per hour. This is because downforce doesn’t have a notable effect until this speed. The formula we use to determine this is shown here. The normal force is just the mass add the downforce. If we put some values in at a speed of 80 metres per second, we get a lateral acceleration of 16.90 metres per second squared. Then if we rearrange the formula for finding the maximum speed around a corner, we can plot the line y equals x squared over the radius of the corner, for the first corner, that’s 59.14 metres. Then we do this for all the other corners and the point where the lines cross is the lateral acceleration for that corner. Now we can go back and put that lateral acceleration into the formula for the maximum speed around a corner.
Now we can move onto the next part of the track, the straight. Now you may remember the values I showed for the straights in the very beginning of the video, these are actually not the true lengths of the straights because we are taking the racing line which cuts of a bit of the straight, the formula for working out this value (x) is here. Let’s put the values in for the end of the first straight and we get 28.97 metres. Just to clarify, that’s how much of the straight is cut off by using the racing line on that corner. Now if we take the values from using this formula on the corners before and after each straight, from the length of the straight shown before, we get these values for the true racing line length of each straight.
Now before we calculate the braking point for each straight, we need the to recalculate the lengths of 2 of the straights, these are the straights where the racing line crosses over from one side of the track to the other. The formula for this is shown here – for the radius of the two arcs that make up the part of the line. Now we put this into the formula for the length of the two arcs and we get 93.29 metres. The only other straight we need to do this for is the second straight, which comes out at 210.97 metres.
Now we have all these values we can find the braking point for each straight. The formula for this looks very, very intimidating so I’m just going to go through it. The braking point (or end of acceleration) equals negative 2 times the braking deceleration times the length of the straight, add the final speed at the end of the straight squared, minus the starting speed at the beginning of the straight squared, all over 2 times the forward acceleration minus 2 times the braking deceleration. If we substitute in the values for the first straight in we get a braking point of 148.18 metres. If you aren’t sure what this means, its just how far along the straight you can go before you have to start braking. Now we have this, we have everything we need to calculate the lap times.
One thing to note is that straight 4, corner 4 and straight 5 are counted as one straight because you don’t need to brake for corner 4. You actually need to start braking for corner 5 during corner 4. To calculate the time spent on each straight, you rearrange this formula here, unfortunately you may recognise this as a quadratic. This means that you can’t use the formula to find time in terms of distance, initial speed and acceleration, unless you use the quadratic formula, which once you simplify it, comes out as this. Substitute in the values for the first straight and you get 4.328 seconds spent accelerating on this first straight. Now to work out the time spent braking, you need to know what the top speed on that straight is, the formula for that is here. Substituting in the values for the first straight we get a top speed of 44.84 metres per second. Then we just use the same formula we used for the acceleration to find that the time spent braking on the first straight is 0.996 seconds. If we do the same for all the other straights, we get these values. And if we add the time spent decelerating to the time spent accelerating, we get the total time spent on each straight.
Now all we need to calculate is the time spent in each corner and for that we need this formula for the length of each corner. If we substitute in the values for the first corner, we get a distance of 92.90 metres. Then we just use the formula here to get 3.174 seconds spent in the first corner. Doing the same for all the other corners we get these values, which when added together give the total time spent in the corners. Then, take the values for the time spent on the straights, and add them together, and lastly add this figure to the total time spent in the corners to get the final lap time of 68.679 seconds.
Over 40 formulas have been considered when producing this video, 4 excel spreadsheets of formulas, all linking and relating to each other needed to produce it, all the animation is my own work, all the modelling of each letter of each formula is my own work. The main resource used when producing this only explained how to calculate the lap times around an oval track, I’ve had to figure out everything else out by myself as there is literally nothing on the internet about this topic.

Bibliography

‘Physics for Gearheads: an introduction to vehicle dynamics, energy and power’ by Randy Beikman Ph.D. (Bentley Publishers 2015)
https://www.carfolio.com/specifications/models/car/?car=510136
http://hpwizard.com/

How to Calculate Time and Distance from Acceleration and Velocity

https://www.desmos.com/scientific
https://www.calculatorsoup.com/calculators/physics/displacement_v_a_t.php
https://www.youtube.com/audiolibrary/music

Attachments
C Jelf calculations SCHOLARS 2019 – excel doc
Scholars project script – word doc