1. The problem statement, all variables and given/known data This is a video of the phenomenon. Start at 5:08 in this video and watch the next two crashes. That first one, where the car hits a wall, straight on, is the most helpful. The phenomenon is that when a car going uphill crashes, it has an upward speed/momentum vector from its path before it crashes (there is a horizontal vector and a vertical vector, in going uphill). Crashing into the wall stops the car from going forward, but doesn't stop the vertical vector, especially in the rear of the car. (The car is long enough that the movement is different in the front and rear. It's not like, say, a marble, where the mass is concentrated in one place. The mass is spread out the length of the car.) In the video, it seems that it doesn't take a huge amount of speed for a car going uphill, to lift off the ground on impact. The questions is, if you know how far you have to jack up the rear of the vehicle to get the rear wheels to leave the ground, and the grade, can you find the minimum speed needed for the vehicle to leave the ground? If the vehicle was braking before impact, the rear wheels would have already started to elevate due to braking. ( See https://en.wikipedia.org/wiki/Weight_transfer and http://www.welltall.com/ymc/discovery/car/wt_xfer.html ) If a car has been braking and hasn't slowed down much, and then hits a wall, can we mathematically model what the minimum speed is they would have to carry, for the rear wheels to lift off the ground? For all 4 wheels to lift off the ground, like in the video? We are primarily looking for an estimated mathematical model with accuracy within about 10% source of error using precalculus math, though a more precise model using more advanced math would also be welcome. The grade progressed steadily from 10% to 1% grade during braking, then the car hit the wall and the rear wheels lifted off and the front wheels just barely lifted off. The distance the car needs to be jacked up, from normal ride height to wheel lift off is 0.3m. The coefficient of friction of tires on road surface is 0.3. 2. Relevant equations basic trig to make vehicle's speed up the hill into horizontal and vertical vectors d= .5at^2+V_0t 3. The attempt at a solution One thing we need to look at is the grade of the road. However, the grade of a road is not usually constant. Would we look at average grade over the course of a certain amount of time back from the collision? Average grade during the time spent braking, before the collision? Grade right at the moment of collision? Grade at start of braking? It seems that the rear wheel lift while braking would be related to the grade throughout the braking process. I am not sure what grade to use, to get the vertical and horizontal vectors. Does what grade to use, depend partly on the friction in the braking process? Better friction means the grade at the start of braking is more important, and worse friction means the grade toward the end of braking is more important? It seems the impact moment, itself, is fairly simple if we understand what happened before that (How much the rear wheels have already lifted before impact, and what the momentum vectors are in the rear of the car before impact...which might have more to do with the grade some distance back, than the current grade). At moment of impact I am using d= -.5at^2+V_0t This is just vertical motion in relation to gravity. V_0 would be the vertical component of the velocity at impact. V_0 would equal delta V because this equation is just looking at from first moment of impact until the car reaches its maximum height, when V=0. So I can substitute at for V_0 on the first round and solve for t, then use the same equation again, plugging in my t, and solve for V_0. I am not sure that's right, though, because when I do this process with higher values for d (which is how far the car lifts off the ground) I seem to get lower values for V_0, which makes no sense. If I can get a model for the collision moment, I can go back and experiment with what to use for grade, and just figure out how much difference it even makes, how I account for grade. It's possible that the choice of how to account for grade makes a small enough difference that it doesn't matter? Are there other dynamics at work, that I am not taking into account?