Elastic Collisions in 1D

I need a nice model to predict the final velocity when two balls collide elastically. Don’t worry why I need this—just trust me.

After working on this for a short bit and making an error, I realized what I need to do. I need to blog about it. A blog is the perfect place to work things out.

So, here is the situation. A ball of mass 10 kg (ball A) is moving with a speed of 0.1 m/s in the positive x-direction. This collides with a 1 kg ball (ball B) moving at 0.1 m/s in the negative x-direction. What is the final velocity of the two balls if the collision is perfectly elastic.

For a perfectly elastic collision, the following two things are true:

  • Momentum is conserved. The total momentum before the collision is equal to the total momentum after the collision.
  • Kinetic energy is conserved. The total kinetic energy is the same before and after the collision.

In one dimension, I can write this as the following two equations. I’m going to drop the “x” notation since you already know it’s in the x-direction. Also, I am going to use A1 for the velocity of A before the collision and A2 for after. Same for ball B.

m_Av_{A1}+m_B v_{B1} = m_Av_{A2}+m_B v_{B2}

\frac{1}{2}m_A v_{A1}^2+\frac{1}{2}m_B v_{B1}^2 = \frac{1}{2}m_A v_{A2}^2+\frac{1}{2}m_B v_{B2}^2

That’s two equations and two unknowns (the two final velocities). Before solving this, I want to find the answer with a numerical calculation.

Numerical Solution

Here’s the basic plan (I’m not going over all the deets).

  • Model the two masses as points with springs on them (not really going to show the springs).
  • When the two masses “overlap” there is a spring force pushing them apart. The strength of this force depends on the amount they overlap.
  • Calculate the position and force on each ball (the force would be the zero-vector in cases where they aren’t “touching”).
  • Update the momentum of the balls.
  • Update the position of the balls.
  • Repeat until you get bored.

Oh, make sure you set your fake spring constant high enough. If it’s too low, the two masses can just pass through each other (which would still be an elastic collision).

Here is what it looks like.

Here is the code (you should take a look). Oh, the final velocities are 0.0636 m/s for ball A and 0.2639 m/s for ball B. Also, here is a plot of the momentum so you can see momentum is conserved.

What about the kinetic energy? Here you go.

Actually, notice that KE is NOT conserved. During the collision there is a decrease in the total KE because of the elastic potential energy. I just thought that was cool.

Analytical Solution

Now let’s get to solving this sucker. I’m going to start with a trick—a trick that I’m pretty sure will work (but not positive). Instead of having the two balls moving towards each other at a speed of 0.1 m/s each, I am going to use the reference frame that has ball B with an initial speed of 0 m/s and ball A with a speed of 0.2 m/s.

Since I am switching reference frames, I am going to rename the velocities. I am going to call ball A velocity C1 and C2 and then ball B will be D1 and D2 (for final and initial). Technically, I should use prime notation – but I think it will just get messy.

So, here is how it looks in the new reference frame.

In general, the initial C velocity would be:


Now I get the following for the momentum and kinetic energy conservation equations.

m_A v_{C1}=m_A v_{C2}+m_B v_{D2}

\frac{1}{2}m_A v_{C1}^2=\frac{1}{2}m_A v_{C2}^2 +\frac{1}{2}m_B v_{D2}^2

Now we have two equations two unknowns. I’m going to cheat. I worked this out on paper and I’m just going to take a picture of it.

Here is the final solution (in case you can’t read it).

v_{C2} = v_{C1}-\frac{m_B}{m_A}v_{D2}

v_{D2} = \frac{2v_{C1}}{\frac{m_B}{m_A}+1}

So, you can get a value for vD2 and then plug that into vC2. After that, you can convert them back to the stationary reference frame to get vA2 and vB2.

Boom. It works. Here is my calculation. Just to be clear, it looks like this:

The output looks like this:

Winning. That agrees with my numerical model.

What is Energy?

I think it is time for me to talk about energy. My ultimate goal is to give some insight into the many stories about perpetual motion. To do this, I will first talk about the fundamentals of energy.

**What is Energy**

I started thinking about this, and at first I realized that I did not have a good, short explanation of energy. The most commonly used definition in science text books is:

*Energy: the ability to do work (or something dreadfully vague like this).*

But what is work? It may be no surprise to find that many college level physics texts avoid defining energy. After some serious contemplation, I think I have this energy figured out.

**There are only two types of energy**

I don’t need a general definition of energy, since there are only two types I can just describe those two. ALL energy is either:

  • Particle Energy: Energy of particles (obviously). I was originally going to just say kinetic energy (energy of things that move) but I forgot about mass (of course you remember E=mc2). This is sort of complicated, so I can perhaps summarize it by saying a particle can have energy because of its mass and because of its motion (really this is just one thing). So, particle energy can be an electron moving, a water molecule moving, or a car (a car is a collection of atoms that are mostly moving in the same direction). For the rotational kinetic energy of the Earth, this is really the same thing. Imagine all the pieces of the Earth (atoms) they are moving and thus have kinetic energy. The idea of rotational kinetic energy is to simplify the calculation. Instead of summing the kinetic energy of each of the atoms of the Earth, one can use the radius, mass, and the angular velocity of the Earth to do the same thing. But realize this is mostly just a short cut.
  • Field Energy: Energy in the fields associated with the fundamental forces – gravity, electric, magnetic, strong nuclear and weak nuclear. Suppose I hold a ball above the Earth, it has particle energy (because of its mass) and there is also energy in the gravitational field associated with the ball and Earth. A chemical battery has energy stored in the electric field due to the configuration of atoms. A final example of energy in fields would be the energy from electromagnetic radiation.

But wait! What about ….. What about …. (insert some energy). All these other energies you read about are one of the above two. Other energies (for example thermal energy) are short cuts. They allow us to deal with large collections of particles without having to calculate ALL the particle energies and the field energies.

**Conservation of Energy**

There have been many many experiments in the history of science. In all of these experiments, the total energy of the situation as been conserved. Well, this is to say that there has not been an experiment where clearly the total energy before something happened was different than the total energy after something happened. Most experiments don’t look at this “energy accounting” directly. Energy conservation isn’t the law, its just what we see. How about a couple of examples of everyday things and I explain where all the energy is?

**Example: A cup of hot tea sitting on a table**

First, where is all the energy in this hot cup of tea? The cup and the tea both have particle energy. The particles (carbon and stuff) have mass energy. If I somehow annihilated this cup and tea it would turn all this mass into field energy. In this case that energy would be in the form of electromagnetic radiation. In fact, this would be so much energy in electromagnetic radiation that it would create pairs of particles (matter and antimatter pairs).

The particles also have energy because of their motion. If we assume the cup is stationary, the particles in the cup are still moving. The hotter something is, the more they move. For the particles that make up the cup, these particles are essentially just vibrating and staying in the same general area. For the tea, the particles are moving around and mostly staying in the cup (but some are leaving at the surface through evaporation). This energy is generally called thermal energy.
The cup also has energy in fields. There is energy associated with the gravitational field of the Earth-Cup(and tea) system. This would be called gravitational potential energy. There is also energy associated with the electric field is the interactions between the electrons and protons in the atoms of both the tea and the cup. People usually call this chemical energy, you could see this energy change forms if you burned the cup or had some other chemical reaction.

As the cup is sitting in the room, it gets cooler. That corresponds to lower particle energies. Where does the energy go? In this case, the stuff surrounding the cup gains energy. The table gets a little warmer (particle energy) and so does the air. This energy transfer takes place by the higher energy particles of the cup and tea interacting (through the electric field) with the particles of the air and the table. You might ask, why is it that the table gains energy and the cup loses energy? Couldn’t it happen the other way and energy would still be conserved? Yes, it would. But the probability of this happening (remember that there are on the order of 1025 particles in this cup) is so near to zero that you have a much greater chance of winning the lottery.
What if the cup were in outer space with nothing touching it? It would still cool (unless the sun was shinning on it). The particles in the cup still radiate electromagnetic energy (usually in the Infra Red region). This IR radiation could causes something else to increase in energy, but the cup still loses energy. The tea would all evaporate and lose energy to IR radiation.

I didn’t think it would be possible to take a simple thing and make it so boring, but I did it. I know that was painful (and likely in some places technically wrong) but it was necessary. Don’t make me do it again. Hopefully, you have an idea of conservation of energy and of the fundamental ideas of energy.

Continue reading “What is Energy?”

Bullets have a lot of kinetic energy (apparently)

I was recently re-watching a MythBusters episode and I found something I had wanted to explore previously (but accidentally deleted the episode). Here is a short clip from the “shooting fish in a barrel” episode:

Did you see what I found interesting? That big barrel of water left the floor from being hit by a bullet.
The question here is: Does a bullet have enough energy to increase the gravitational potential energy of the barrel to that height?

Continue reading “Bullets have a lot of kinetic energy (apparently)”