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hahahahahahah oh my god
Ok so i did some math and this is my conclusion
Ok. Time for Tracker Video Analysis. Here is the y-motion for the ‘flight’:
Notes:
Now, if I assume that the acceleration in real life is 9.8 m/s2, then the ramp would be 2.45 meters tall. Using this new scale, I can look at the horizontal motion:
This looks linear-ish. From this, the horizontal velocity is mostly constant at a value of around 16 m/s (which is about 36 mph for the metric-challenged). At least I have enough info to make some calculations. Note that 16 m/s is the guy’s horizontal velocity, not the total initial velocity. The initial vertical speed can be determined by looking at the time in the air. (here is a review of projectile motion) If I assume that the guy starts and lands at the same height, then I can use:
Since y and y0 are the same, I can solve for the initial velocity:
From the video, ?t = 2.1 seconds. This gives an initial y-velocity of 10.3 m/s. This will give a total initial speed of:
Putting in the values for the x- and y-velocity, this gives a magnitude of the initial velocity of 19 m/s. Why do I care about this velocity? Two reasons. First, I can estimate how high up the hill the guy would need to start to get this speed. Second, this is the same speed the guy will hit the pool. So, I can estimate the acceleration when he lands and see how deadly it would be (I already suspect he should be ok – think about professor splash)
How high up the hill would he have to start? If I ignore friction (always a good place to start), then I can use the work-energy principle to calculate this. Let me make a sketch.
The work-energy principle is great to use here because it essentially deals with change in position. I will start with the Earth-guy as my system (this means that there will be a gravitational potential energy and NOT work done by the gravitational force). When working with the work-energy principle, you need two positions. In this case, that will be at the top of the hill and at the top of the ramp. During this motion, there are only two forces acting on the guy: the normal force from the ground and the gravitational force. The normal force does no work since it is always perpendicular to the direction the guy is moving. Gravity doesn’t do any work because I am using the gravitational potential energy. If the guy starts from rest at the top of the hill, and I set the zero gravitational potential at the top of the ramp, then:
I didn’t want to be too confusing about the velocity in the above expressions. That is the velocity at the top of the ramp. If I wanted to be consistent with the stuff from before, this would be v0. Using this stuff and solving the for the height above the ramp, I get:
Notice that this solution does not depend on the mass of the guy nor does it depend on the angle the hill is inclined. If I plug in the value for the speed at the top of the ramp, then the starting point must be at least 18 meters higher than the top of the ramp. If there is significant friction it would need to be even higher.
It is very difficult to estimate the height of the starting point because of the angle the camera is viewing from. There is one thing that does not change with perspective though – time. I can get the time it takes the guy to get from the top of the hill to the bottom of the hill and calculate how steep the hill would have to be (again assuming no friction). From the video, this is about 3 seconds. The ramp looks pretty big, but I am going to use the velocity at the top of the ramp as though it were the velocity at the bottom of the ramp just to get an estimate of the angle of the ramp. Ok, so if he goes from 0 to 19 m/s in 3 seconds, then his acceleration (average) would be:
So, if this were a hill at a constant slope with no friction, how steep would it be? Here is a free body diagram of an object sliding down a slope.
I want to find the acceleration down the plane as a function of the angle of the plane. In this case, the only force acting in the direction of acceleration would be a component of the gravitational force. This gives:
If I put in 6 m/s2 in for a, then I get an angle of 40 degrees. Pretty steep – but it is a mountain I guess. I guess this is real. But there are still some things to investigate. I will leave the following questions for homework:
Homework hint. If you look at that Professor Splash jumping into a foot of water, it will really help. In that analysis, Prof Splash is going about 15 m/s before hitting the water. Yes, that is slower than this guy, but this guy lands in much deeper water (maybe 3 feet?) and at a non-perpendicular angle (which means he has a greater distance to slow down).
