A couple of days ago I noticed a blog post from Chad Orzel entitled How Deep Does Veritasium’s Bullet Go?. It links to a video by Veritasium that asks which block goes higher, one shot with a bullet on center, or one shot with a bullet off-center. Take a look at the video.
Before you go any further make an educated guess as to which block goes higher. Rhett Allain at Dot Physics
goes through an analysis of what happens here
. Since Chad and Rhett did a pretty good job covering the physics I want to talk about this from a broader perspective. Most of the answers I saw on different websites talked about energy conservation and only a couple actually mentioned momentum conservation. I’m not completely sure why energy dominated the conversation, but I bet it has to do with the fact that more time is spent on energy than momentum in most physics curricula, and people seem more comfortable with energy, even though I think most students have a stronger conceptual grasp of momentum than of energy.
is defined as the mass
of an object multiplied by the velocity
of the object (for this post we’ll assume we are dealing with a classical system moving at much less than the speed of light) or
The arrow over the
mean that both of these are vectors. A vector contains two pieces of information; how much, and which way. A velocity vector tells you how fast the object moves and which way. The momentum vector tells you how much momentum the object has and which way the momentum is pointing. Energy, on the other hand, is a scalar quantity, which only provides information about how much, and has no directional component to it. The kinetic energy
of an object is defined as one half of the object’s mass
times the magnitude of the velocity
. Notice than none of the terms in this equation are vectors (the magnitude of the velocity, or speed, is required, rather than the velocity, which is a vector).
The fact that energy is a scalar quantity makes it much easier to use than a vector equation. If you’ve ever dealt with vectors you will remember that you need to keep track of three pieces of information, instead of the single number needed for a scalar. The fact that there is no direction associated with energy which simplifies the math considerably. However, to work with energy you have to throw out that piece of information that tells you “which way” which means you loose some idea about what your system will do. Momentum requires you to keep track of more information, which means more calculations are needed.
Here is a simple example of the limitations of energy. If I know the speed of a ball that is launched in the air I can use energy conservation to calculate the maximum height the ball could reach. However, if the ball is launched at a
this calculation won’t tell me how high the ball actually goes without a few extra calculations. If I used momentum instead, I could figure out how high the ball goes, where it reaches its peak height, where it will land, and how long it was in the air. Of course this will take a few more calculations to do, but all of the relevant information is retained.
The information loss yields another important (potential) limitation on energy. Since energy does not conserve information about the direction of motion associated with energy, whenever two objects collide, like the bullet and block in this problem, the kinetic energy is able to split up into different directions. In particular, collisions can cause the macroscopic kinetic energy of the incoming bullet to be transformed into internal energy of the system, either by heating things up or deforming the objects. This means that it can look like energy seems to disappear altogether. The energy is still there, it is conserved, but it is hidden from our eyes inside the block.
Momentum, with its directional nature, is much harder to hide (but not impossible*). This means that however much upward momentum the system originally has (i.e. the bullet shot upwards) is exactly how much momentum we will be able to see after the collision (i.e. the block with the bullet embedded). Since the bullet in both cases has the same initial upward momentum, both blocks after the collision will have the same final momentum, which means that must reach the same maximum height.
Of course now you are asking “what about the spinning, doesn’t that require momentum?” Spinning does require momentum, but a different flavor of momentum; angular momentum. What we have been talking about up until now was linear momentum or translational momentum. It turns out that angular momentum is also conserved during these collisions, but it isn’t quite as obvious. It doesn’t make sense to talk about the angular momentum of the system. You have to specify a reference point, so we’d rather talk about the angular momentum of the system about the center of the block**. When the bullet is fired straight towards the center of the block, the bullet doesn’t have any angular momentum about the center of the block (imagine running straight towards the center pole of a merry-go-round; when you hit the merry-go-round you wouldn’t make it spin at all). When the bullet is fired off-center, it does start with angular momentum about the center of the block (now imagine running towards a merry-go-round slightly off-center and grabbing on; this would cause the merry-go-round to start to spin). The reason you see the block spinning after being hit by the bullet in the off-center case is because angular momentum is conserved, and the bullet started with angular momentum about the center of the block as it was shot from the gun.
So what is the take-away message from all of this? Energy, while useful, has its limitations. You throw away useful information about the system when you start thinking in terms of energy. Energy will help you figure out the limits of, say, how high the system might go, but it won’t always tell you exactly what will happen. Momentum will give you a more complete view of what is really going on and is more likely to allow you to determine exactly how the system will move. Be careful when using energy and make sure you are aware of the limitations of your tools.
* It is left as an exercise for the reader (or a future blog post) to figure out how momentum might seem to disappear from view
** It’s actually about the center of mass of the wood block and bullet system, but given the low mass of the bullet these two points are pretty close together.