Tesla Model-S is a great car

              So I was reading recently this article on the website called electrek.  This site is dedicated to publishing news on electric vehicles.  In the article that is mentioned above there is a story about a girl that takes her fathers Tesla and then crashes it.  The cool thing about this story and why it is not a tragic story about 5 teenagers deaths is that the Model-S is such a safe vehicle.  Oh and did I mention that before the crash the car was in the air for 82 feet.  I thought that this would be an interesting projectile motion problem to find out how fast the car was going in order to be in the air for 82 feet.
So here we go...


So lets start with a few assumptions.
1. The car didn't go off any cliffs so our starting altitude and our final altitude are the same.
2. The edge of the road was a ramp (most highways are banked)
3. There was no air resistance when the car was in the air.
4. The car was going faster than 20 m/s (about 45 mph)
5. There was not enough wind to need to worry about it.

            Lets talk about our assumptions and tell why I think they are pretty good.  First in the pictures of the car crash that I saw in the article there was no cliffs and the area looked to be farm country. So I think that our first assumption is a pretty good one.  Second most if not all highways are banked. This gives more control to the drivers of course that didn't really help in this situation but usually this helps.  If you want a better explanation of why it helps you can always go to this Wikipedia article.  Which I think is pretty well written and has sound physics.  So our second assumption is good.  Third the Tesla Model-S is one of the slipperiest car on the road today.  It has a drag coefficient of 0.24 which is one of the best.  So our assumption that there was no air resistance when the car was in the air is not a very good one given that it was probably moving really fast when it made this jump but because the Tesla is so slippery I think we can still assume this and get a pretty accurate picture of what happened.

          I think the best way to solve this type of problem is to do computer model.  Basically I am going to write some code for a computer to run that is a bunch of guesses and see what one is the closest.  So here is the program....




          While you look at the code you will notice that you can also change it.  So if you would like to mess around with it and see what you can get be my guest.  Before you do though you can see that I tried to account for every situation where the car would go about 25 meters through the air. Also if you look at the code on line 17 you can see that I left the air resistance in there. I did this mostly because I found all the information I needed to find the air resistance of a Model-S here.  If it was front facing the whole time.  I am some what confident that it was not forward facing the whole time it was in the air.  I imagine that there was some twisting and turning that didn't always have the front of the car facing where the car was going but I am willing to bet that it was pretty close.  So I guess we have replaced one of our assumptions with another one but I think that this one will be a little bit more accurate to the real world.   If you run the code by hitting the play button at the top of the code then you will see there are about 5 situations where the car goes a distance close to 25 meters. I am sure that if we wanted to be more accurate we could change our angle by the tenth of an angle and/or our speed by tenth of a m/s but I think this will be accurate enough for our needs. All our situations have angles between 5 and 10 degrees and speeds between 27 and 38 m/s.  Now that is not a huge difference in angle but it is a pretty big difference in speed. So lets try to narrow it down a little bit.

          I think the best way to narrow it down would be to narrow the angle.  Now like I said a 5 degree angle is not the big but when you are talking about the angle of a highway it is a pretty big difference on how they angle the road.  You may have seen those signs that tell you to go a certain speed around a turn.  like this one..



Have you ever wondered where they get the speed from. well you are now about to find out.  You see it is all about physics.  So the civil engineers that design the road bank the road at a certain angle based on the speed and the radius of curvature that they want that road to be.  Basically it comes down to a free-body diagram and summing forces.   The Wikipedia article that I mentioned earlier about banked curves is pretty good so I will borrow their free-body diagram.



As you can see you really just sum all the forces in the x direction which gives you



Then you sum the forces in the y direction and you get.



Solving for the normal force and substituting in equation into the other and then solving for the angle you get...







which is the equation that they would use to determine the angle at which to build the road.  Now if we plug in a speed of about 20 m/s (45 mph)  this website says that there should be a minimum radius of curvature of 196 m (643 ft) so we plug all that in and we get an angle of  19 degrees.  Which was one of the angles that was in our simulation that corresponded to a speed of 20 m/s.  Which is a little disappointing in that I thought that since they traveled 82 meters in the air that they would have been going much faster.  But if my calculations are correct the only thing that they did was forget to turn. They were not speeding all that fast because when the left the ground at 19 degrees they were only going what seems to me is the speed limit.  There you go not an exciting end but there is the conclusion  that I came up with. I guess this is one of the very few times where science isn't that exciting but it still is an interesting result.