Inspired by the famous scene in the movie, Indiana Jones and the Raiders of the Lost Arc, we set about to discover the optimal slope needed to slide a mass from the top of a hill to the position of a hapless grave robber, while taking drag and friction into account. We found that a slope of -1.79 gets the mass in question from point A to point B in 1.97 seconds, faster than any other.
In order to protect the secrets hidden within the ancient crypts, you are building a trap that involves a large mass of stone sliding down an inclined plane, and towards a hapless intruder. Assuming that point A, the point from which the block is dropped, is 10 meters above the ori gin, and point B, the position of the intruder, is 10 me ters to the right, what slope of the plane gets the object from point A to point B the fastest?
As the brick itself is the moving part in question, many components of the system depend on the traits we as signed to it. For the purpose of the simulation, we gave the brick dimensions of a 0.15m cube, and assigned it a mass of 6.48 kg. We found that the coefficient of drag on a cube is 0.8, and the coefficient of friction between the brick and the ground, which we decided was made of wood, is 0.6.
In order to simulate this model, we defined a few key points. The force of gravity was determined with the gravitational constant (g) and he slope of the hill (θ). Friction was calculated using gravity, the slope of the hill and the coefficient of friction (μ). Drag was determined by the density of air (ρ), object velocity (v), coefficient of drag (Cd), and the surface area of a single side (A).
In order to make sure our model worked accuratly, we first tracked the position of the brick, without friction and drag, as it traveled down a single de fined slope.
Once we had gotten a reasonable output we com pared it against hand calculations to make sure our model matched the actual physics. We then added in drag and friction while making sure the results remained plausible. We then manually tested a few points between -5 and -1 before finally sweeping through all slopes within our previous boudaries.
After sweeping through possible slopes, we plotted our results against the duration of the time it took for the brick to travel from point A to point B. We found the op timal slope of a brick cube traveling down a wooden in cline plane to be -1.79. Using this slope it took the brick 1.97 seconds to travel down 10m and right 10m. We found that the shortest past is not nessesarily the fastest when dealing with gravity, friction and drag.
The real world is never perfect, and because of this, we must make a few generalizations about the situation.
- The simulation uses a single, constant coefficient of friction.
- While the situation we are modelling likely has a curve, our model does not. We account for this by ensuring that velocity is conserved from the slope to the hori zontal plane.
- We are modeling a situation where the brick in question is perfectly flat, with no surface imperfections.
- We set the air density and force of gravity equal to that at sea level.
- We are using rough approximations for the drag coeffi cient of a cube, and for the coefficient of friction for brick on wood.
- We only model two dimensions, removing the chance of moving along the z-axis, as a three-dimensional object would.
- As the results are only tested with equivalent rise and run, the optimal slope could be different under different circumstances.
- The model fails to account for variations in the coefficient of friction that occur as material, humidity, and other similar variables change.
Though our simulation came up with an answer, there are many possible improvements we could make, including:
- Using a round object rolling down a linear ramp, which would involve including rotational physics
- Finding the optimal curve between two points
- Including a person running away from the sliding brick to bring our model closer to reality
- Jupyter Notebook
- Python 3
- Kyle Bertram
- Khang Vu
- Mika Notermann
- ModSimPy by Allen Downey