Why did the window glass break on the Tesla Cybertruck?
The Tesla “Armor Glass” on the Cybertruck broke because it was never tested properly and the team did not understand the difference between elastic and inelastic collisions, and the impact of rigidly constraining a body.
Oh my fucking god.
That was the response of Elon Musk, CEO of Tesla, after Franz von Holzhausen broke not one, but two, windows made of “Tesla Armor Glass” at the reveal of the new Tesla Cybertruck on Thursday night. But why did it break? The answer does not lie in the materials used, but in your Physics 101 class. Do you remember elastic and inelastic collisions?
In short, a collision between two objects is considered to be “elastic” if most of the energy is maintained in objects colliding, and inelastic otherwise. For instance, if I hit one billiard ball with another, the pair will be traveling at about the same speed after the collision as before (elastic). If I hit a billiard ball against foam, most of the energy will be absorbed and it will be traveling slower after the collision (inelastic).
Another factor to remember is rigidity. In short, if you hit something that does not deflect (steel, glass, etc.), then the contact time is short. If you hit something that does deflect (foam, skin, etc.), then the contact time is longer. The longer the contact time, the less energy per unit time is transferred. Generally, less energy per unit time is good and helps to avoid material failure (such as shattered glass).
Time for some math.
First, what is the frame rate for the video Elon tweeted of the test?
Elon tweeted a video showing the pre-test of the Tesla Armor Glass and supposedly proving that it can withstand being hit by a metal ball. The below two screenshots show the location of the metal ball that was thrown at the Armor Glass after it bounces off at times t1 and t2 (both are on rebound).
In between these two frames, the ball is under the force of gravity and thus the vertical distance it falls can be approximated as:
d = 1/2 * g * t^2
Assuming that Franz is an average height male (for reference), we can assume that the distance the ball has fallen is ~3ft. The acceleration due to gravity is a constant, 9.81 m/s^2. This yields a time of 0.43 seconds between the frames. These two images are 85 frames apart, which comes to just around 200 frames per second (fps) for the video which is reasonable for a low cost high speed camera.
Second, what is the momentum change of the metal ball after it hit the Armor Glass in the pre-test?
Assume the steel ball has a diameter of 3 inches. This is a volume of 14.1 in3. Assuming an average steel density of 7.9 g/cm3, this yields a weight of 4.0 lb.
(For simplicity we are only considering the horizontal component of velocity.)
Let’s assume that Franz’s hand is 4.9ft from the Cybertruck when he releases the ball (this is estimated from the video). It takes 45 frames for the ball to make contact with the window. That is 0.225 seconds (@ 200 fps). We can give the ball a constant velocity as it travels from his hand to the window, and thus the ball has a speed of 22.2 ft/s. This yields a kinetic energy of 41.5 Joules.
For the bounce back, the ball travels 4.83ft away from the Cybertruck over 0.54 seconds (108 frames). This yields a speed of 8.9 ft/s and a kinetic energy of 6.7 Joules. It is important to remember (as Elon often says when talking about rockets re-entering the atmosphere) that energy is proportional to velocity squared, so as the speed decreases, the energy decreases at a faster rate.
So here we have the incoming energy of the metal ball at 41.5 Joules and the rebound energy at 6.7 Joules. That means the window absorbed about 85% of the total energy of the metal ball, and most of it is taken up by the flexing of the window, which occurred over ~2 frames or about 0.01 seconds.
So what happened during the Cybertruck reveal?
It is a little harder to see exactly what happened during the truck reveal when the glass broke (twice). This is because the camera is not high speed, and the angle makes it much harder to estimate distance. But, it appears that the ball drops to the ground about halfway between Franz and the truck. We can estimate this is about 2.5 ft.
If this is the case, then we can assume that much more than 85% of the energy of the ball was absorbed by the window. In fact, it was probably closer to 95% (or more) of the energy that was absorbed. This, and the inability of the window to flex, is what caused it to shatter.
But more important than the total amount of energy that was absorbed (or the total change in momentum of the metal ball) is the rate at which the momentum changes. The rate of momentum change is called a force. When the glass is allowed to deflect, it makes contact with the ball for a longer period of time. This means that the ~35 Joules can be absorbed over 0.01 seconds. When the glass is rigid, the ~35 Joules are absorbed over 0.001 seconds which is an order of magnitude higher impact force. (Those time values are just estimates because the video frame rates are too slow to actually estimate the impact time.)
The equation for force is:
F = m*(v1 — v2) / t
Here m is the weight of the ball, v1 and v2 are the speeds before and after the collision, and t is the time the ball was in contact with the window.
If the absolute value of the force experienced by the window pane is higher (both because the total momentum change is higher and the contact time is lower), then it will be more likely to experience a failure.
What about the other drop test?
Right before throwing the ball at the truck window, Elon did an on-stage drop test where a metal ball was dropped directly onto a pane of glass. This glass pane was also not rigidly constrained (just a few clamps), so it was able to flex more and thus not shatter.
So in summary, why did the glass break?
In the pre-release test, the glass was in constrained. This means the collision is more elastic (the metal ball maintains ~15% of its kinetic energy after impact) and the contact time between the ball and the window is higher. The resulting force on the window is around 1700 Newtons.
On stage, the window is fully constrained by the Cybertruck. This means the collision is less elastic (the ball drops straight to the ground and only maintains ~5% if it’s kinetic energy after impact). The impact itself is also faster so there is less contact time between the window and the ball. The resulting force on the window is around 14,000 Newtons, nearly 10x more!
What is the takeaway for Elon and the Tesla team?
The main takeaway from this snafu is to “test like you fly.” In short, if a glass window is going to be used in a certain way, it should be tested in that exact way. Just because the glass does not break when you perform a test in a lab (not part of a system), does not mean it will survive when it is rigidly secured inside of a window pane.
This applies to myriad other situations where it is critical to ensure that the test situation is appropriate for the use case. From aerospace to autonomous vehicles (to windows), testing must be representative otherwise it is pointless. Even though Elon said the glass was tested before, the test was not valid.