Ok so i did some math and this is my conclusion
- What is the guy’s acceleration after he leaves the ramp?
- What was his initial velocity leaving the ramp?
- How high above this would he have to start?
Ok. Time for
Tracker Video Analysis. Here is the y-motion for the ‘flight’:
Notes:
- The unit of scale is the height of the ramp.
- There
is an obvious perspective problem. As the camera pans, the angle
between the motion and the camera is not constant and not
perpendicular. I have no simple way to correct for this (yet).
- From this, the acceleration is about 4 ramps/s2.
- At
this point, I can’t really tell if it is fake or not. The motion does
not fit a parabola very well, but that could be because of perspective
issues.
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:
- Suppose you are planning this “stunt” and your initial velocity
is off by dv (some small amount). What would the resulting change in
range be? If dv = 0.5 m/s, would the guy still land in the pool?
- Suppose
the coefficient of kinetic friction was 0.1. What would be the new
velocity at the top of the ramp? You can assume that the hill is
straight.
- Estimate the acceleration of the guy when he hits the water. Look up the NASA g-force tolerance tables and see if he is ok.
- Where did they get all the water to fill up the pool?
- Who inflated the pool and how long did it take if they just used their lungs?
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).