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Minecart Kinematics
By Wyboth
Abstract
The purpose of this experiment is to explore the physics surrounding minecarts. I did experiments to test if collisions are elastic or inelastic, to test if momentum is conserved in collisions, and to determine the coefficient of kinetic friction for minecarts.
Introduction
One of the first things students do in physics class is roll carts down tracks to learn about energy and momentum. This experiment will do the same, but with minecarts. To determine how similar or different minecart physics are from real life, I will conduct several experiments with minecarts. I will test to see if collisions are elastic or inelastic, to determine if our current momentum laws are valid in Minecraft, and to determine the coefficient of kinetic friction for minecarts.
Materials and Methods
The first experiment will determine whether minecart collisions are elastic or inelastic. An elastic collision is a collision in which the two objects bounce apart after colliding, like a rubber ball bouncing off of a wall. An inelastic collision is the opposite, the two objects stick together, like play-doh sticking to a wall after colliding. To determine this, I placed a minecart above a sloped track, and a minecart at the bottom of the sloped track. Because occupied minecarts behave differently than unoccupied minecarts, I placed a zombie in each minecart to make them occupied. I recorded the experiment. You may watch the video here (http://youtu.be/5LzYXv3kFjI). As is evident in the video, the two carts stick together after colliding. This means that the collisions are inelastic.
The next experiment will test if our momentum laws are true for Minecraft. In inelastic collisions, momentum (p) is conserved. This means that:
p1 + p2 = p3
Or the momentum of the carts after they collide is equal to the momentum of the first cart plus the momentum of the second cart before collision. Momentum is defined as mass times velocity. This relationship is shown in the equation for momentum:
p = m * v
So, the first equation can be rewritten like this:
m1 * v1 + m2 * v2 = m3 * v3
Now, in Minecraft, it is impossible to measure the mass of an occupied minecart, at least in conventional units. However, it is not necessary to do so. As long as both minecarts have the same mass, it is simply a matter of ratios. I will give each cart a mass of 1 minecart (1mc). To prove if the above equation works, I will perform a similar experiment to the first one, but with myself in the minecart above the slope in order to measure the velocity of the minecart before and after the collision. The other minecart will contain a zombie. Because the minecarts do not contain the same entity, it is questionable if they have the same mass. I tested this by setting up two minecart tracks similar to the track from the first experiment side by side. I put a minecart containing a zombie at the top of the hill and another minecart containing myself at the top of the hill on the other track. I positioned 2 pistons behind the minecarts connected to a button in order that both minecarts are pushed off at exactly the same time with exactly the same velocity. If the two minecarts have the same mass, they will travel along the track at identical speeds. I recorded this experiment. You may watch the video here (http://youtu.be/o8y3YTM4Uho). As the video shows, the zombie and I traveled at exactly the same velocity down the track. This means that both minecarts had the same mass.
Now that I have determined that a minecart with a zombie and a minecart with a human have the same mass, I can conduct the actual experiment. I will determine the minecart's velocity the instant before collision and 1 second after collision by using F3 and frame analysis. The minecart's velocity before impact was 10.308 m/s and the velocity after impact was 4.8741 m/s. I will solve for v3 in the momentum equation and compare that velocity to the recorded velocity.
m1 * v1 + m2 * v2 = m3 * v3
1 mc * 10.308 m/s + 1 mc * 0 m/s = 2 mc * v3
10.308 m/s = 2v3
v3 = 5.154 m/s
5.154 m/s seems very close to 4.8741 m/s. I will calculate percent error.
(abs(4.8741 - 5.154) / 5.154) * 100 = 5.43%
A 5.43% error is very reasonable. Because there was a small percent error between the theoretical velocity and the actual velocity, I can conclude that the law of conservation of momentum does apply to minecarts in Minecraft.
The final experiment will be to determine the coefficient of kinetic friction between the minecart and the rail. The coefficient of kinetic friction determines the force of friction on the minecart. The force of friction is equal to the coefficient of kinetic friction times the normal force, which is the force exerted by the ground on the minecart. The formula for coefficient of kinetic friction is:
µk = Fµ / Fn
In this equation, µk is the coefficient of kinetic friction, Fµ is the force of friction, and Fn is the normal force. Since Fµ and Fn are both forces, and since F = m * a, the equation can be rewritten as:
µk = (m * aµ)/(m * ag)
In this equation, aµ is the acceleration caused by friction, and ag is the acceleration caused by gravity, and m is the mass. Since mass is divided by mass, they cancel, and leave us:
µk = aµ/ag
In my last paper, Mass of the Minecraft Earth (http://www.opticraft.net/index.php/topic,16472.0.html), I determined that the acceleration due to gravity was 14.4622 m/s2. Thus, the equation now becomes:
µk = aµ/14.4622 m/s2
The only unknown besides µk is aµ. This can be found by comparing the velocities of a minecart 1 second apart. I chose a part of my video where there was an uninterrupted second of traveling over the flat track. At the point I chose, the minecart had a velocity of 7.2729 m/s. 1 second later, the minecart had a velocity of 7.1271 m/s. This equation can be used to calculate acceleration:
vf = vo + a * t
In this equation, vf is final velocity, vo is original velocity, a is acceleration, and t is time. I will substitute the known values into the equation.
7.1271 m/s = 7.2729 m/s + a * 1 s
7.1271 m/s = 7.2729 m/s + a
a = -0.1458 m/s2
Now that we have aµ, we can substitute this into our equation.
µk = -0.1458 / 14.4622
µk = -0.010081453721
I would note that µk is a scalar quantity, so I will drop the negative sign and round to a more reasonable value.
µk = 0.01
This means that to calculate the force of kinetic friction on a minecart, you can use:
Fµ = 0.01 * Fn
Results
The results of these experiments are that occupied minecarts have inelastic collisions, the law of conservation of momentum is true for inelastic collisions between occupied minecarts, and the coefficient of kinetic friction for an occupied minecart is 0.01.
Discussion
Firstly, I will cover sources of error. There is only one source, and that is inexact frame capture. Basically, my recording software probably did not capture frames exactly when events occured. You can see the effect of this if the difference between theoretical velocity and actual velocity in the conservation of momentum experiment. This would have affected the conservation of momentum experiment and the coefficient of kinetic friction experiment. This is unavoidable.
Now, I will discuss the results. I was surprised to learn that minecarts have inelastic collisions. I was expecting for them to have elastic collisions, as most "carts" do in the real world. If they did have an elastic collision, the cart pushed off the hill would have stopped, and the other cart would have shot off at the same speed that the first cart was going before the collision, just like a Newton's cradle. However, they very clearly stuck together, so the collision was inelastic. I was also excited to find out that conservation of momentum was true for Minecraft. While recording the video, I did not notice the drop in velocity, so I was expecting it to not apply. I was shocked and delighted to discover that the velocity had been halved.
There are no conclusions that can be drawn from these results other than those already found. This paper is more of a report of findings than a study to be ruminated on, as my last paper was. However, these findings will be extremely useful for future calculations involving minecarts.
Works Cited
Minecraft, developed by the Mojang Team
Newton's Second Law of Motion
Law of Conservation of Momentum, Isaac Newton
Equation for Coefficient of Kinetic Friction, Arthur-Jules Morin