Have you ever heard of the Newton's Cradle? It's that one physics "toy" that many executives and businessmen have lying on their office desks, with metal balls hitting each other. Yeah, you know what I'm talking about. Even if you didn't, here's an interactive demonstration anyways. You can move the balls around, but try not to break anything. (Not that it really matters if you do, there's a reset button.)
Newton's Cradle Demo
Now, what does this have to do with momentum? Well, notice that the number of balls set in motion initially is equal to the number of balls that "pop up" after they collide. And, at least at first, the balls seem to fly away with equal velocities. The physics engine isn't perfect, so eventually the behavior decomposes.
This indicates that the momentum carried by the balls coming in is the same as the momentum carried by the balls leaving! This leads us to a new physics principle: the conservation of linear momentum.
Like the conservation of energy principle, there are certain conditions that must be satisfied for momentum to be conserved. The conditions are very similar, but a little less restrictive. The total linear momentum of a system will be conserved if no net external force acts on the system. A key takeaway is that like energy, momentum is conserved for a system of objects, not typically just a single object.
You might recall from our discussion of energy that mechanical energy is not conserved if there are nonconservative interactions. Well, momentum simply does not care about this. Internal forces, be they conservative or not, will still result in conservation of linear momentum. In fact, we'll learn about cases where momentum is conserved but mechanical energy is not.
We can use the idea of impulse to explain why internal interactions conserve momentum. According to Newton's Third Law, if one object exerts an impulse on another, it will receive an oppositely directed and equal impulse from that object. This means the total change in momentum of the two objects as a system will be zero! Now, we said before that mechanical energy isn't always conserved. The same is true for linear momentum! The conditions for conservation of linear momentum are actually more general than the conditions for conservation of mechanical energy, if you can believe it. However, the way we think of momentum conservation is slightly different than energy.
First off, what do we mean by conservation of momentum? What are the objects we're going to deal with? Well, we're almost always dealing with a system of objects. Think about two balls hitting each other. Neither ball's momentum is individually conserved, but their sum, which is equal to the total linear momentum of the system, will be conserved after the interaction.
There is another key idea that is associated with conservation of linear momentum. If all of the previous conditions are satisfied, we can conclude that another fact must be true. I'm going to introduce the idea of the center of mass, which can be thought of as where a system will "balance" under the influence of gravity. It's really the normalized form of the first moment of mass, however:
$$ x_{cm} = \dfrac{m_1 x_1 + m_2 x_2 + m_3 x_3 + ...}{m_1 + m_2 + m_3 + ...} $$ This can be written as a summation, and I think you could do it just by analyzing this equation. Because we're trying to go over this concept for the sake of momentum, I won't write it here. The center of mass velocity and acceleration are defined similarly.
$$ v_{cm} = \dfrac{m_1 v_1 + m_2 v_2 + m_3 v_3 + ...}{m_1 + m_2 + m_3 + ...} $$ $$ a_{cm} = \dfrac{m_1 a_1 + m_2 a_2 + m_3 a_3 + ...}{m_1 + m_2 + m_3 + ...} $$ For the condition of no net external force on a system, the center of mass acceleration will be zero (if you generalize Newton's Second Law to entire systems, you will get this result). This means that the center of mass velocity will remain constant while momentum is conserved. We will talk about the center of mass in detail in a future lesson.
The "center of mass" is really an abstract concept that isn't tangible. However, if we draw in the center of mass on a diagram and calculate how it moves while bodies in a system are interacting with each other, we will see that even if the motion of the bodies in the system is very chaotic, the center of mass remains moving in a straight line with constant velocity. If you don't understand this now, don't worry.
With that little detour out of the way, we can finally express the Conservation of Linear Momentum in words: With that, we can talk about the conditions under which linear momentum will be conserved. It's actually a very simple and elegant condition: linear momentum is conserved if no net external force acts on the system. This might sound very similar to the condition for mechanical energy conservation, but with one key difference.
Mechanical energy is only conserved if there are no nonconservative internal forces. In other words, if the objects in the system have friction with each other, mechanical energy won't be conserved. However, momentum is always conserved as long as there is no net external force. Even nonconservative internal interactions will preserve the total momentum.
Our old friend Newton's Third Law comes in here to help explain why. Newton's Third Law states that a force will have an equal an opposite reaction force, which means that an impulse exerted by object A on object B will result in an equal and opposite impulse exerted from object B on object A. This means the total change in momentum of the system balances out to be zero, since the two impulses are equal and oppositely directed! See, the old concepts we learned such a long time ago are still coming to haunt — I mean help — us!
We can finally write the sentence that codifies the conservation of linear momentum:
The Law of Conservation of Linear Momentum: If no net external force acts on a system, the total momentum of the system must remain constant.
This might not seem useful right now, but in some specific scenarios it is essential to use this result in order to solve problems. By itself, this conservation law actually cannot really solve problems, but given just a few more initial conditions and/or constraints, we can unleash the full potential of this new conservation law.
We will be talking more in-depth about the two most common types of momentum problems, but here I'm going to present a very generalized problem on momentum just to get you started.
An astronaut floating in outer space has a mass $ M = 60 ~\textrm{kg} $ and is carrying a box of tools of mass $m = 15 ~\textrm{kg}$. She has run out of propulsion power on her backpack and must return to her spacecraft by other means. First, how would she achieve this?
Figure 1: The astronaut's predicament. Well, in the vacuum of outer space there are no forces acting on the astronaut. Therefore, the total momentum of the system has to be conserved. So, if one of the bodies was to be given some momentum, the other one would also gain an equal amount of momentum in the other direction!
This is the key: she must throw the box of tools away from the spacecraft such that she gains some momentum and begins to drift towards her spaceship. Like I mentioned in the last lesson, this is also sort of how rockets work.
Now that we have the mechanism of action established, let's add in a few more factors so we can make this a solvable problem.
The astronaut, unfortunately, was not conservative with her oxygen and now only has 5 minutes worth of it left. She is stranded a distance of 3 kilometers from the spaceship. How fast must she throw the box of tools such that she can make it back in time? She is stationary relative to the ship.
First, we can do the easy step and find the required velocity for the astronaut to just be able to drift back in time. We know the amount of time she has left as well as the distance to the spacecraft, so this is an easy kinematics calculation.
$$ v_{a} = \dfrac{x}{t} = 10 ~\textrm{m/s} $$ Next, we can use this velocity along with the fact that total linear momentum must be conserved in order to solve the problem.
$$ p_{tot} = 0 $$ $$ Mv_{a} = m v_{b} $$ The subscripts $a$ and $b$ denote the astronaut and box, respectively. We are able to write the second equation because the two momenta are directly oppositely directed (momentum is a vector!). With this, we can solve the problem:
$$ v_b = \dfrac{M}{m} v_a = \bbox[3px, border: 0.5px solid white] {40 ~\textrm{m/s} } $$ This is around 89.4 miles per hour. We better hope our astronaut hits the gym frequently. An astronaut is stranded in outer space after her propulsion runs out. She is only carrying a box of tools and nothing else. Is there any way she could return to the spacecraft before her oxygen runs out?
Figure 1: The astronaut's predicament. First off, you might think that just flailing your arms around (in a methodical way) might get you moving. After all, this is what swimmers do, and even skydivers can somewhat control their movement and maneuver in the air this way. However, space is a vacuum where there is literally nothing.
On Earth, you can push against water or even air because they have some mass, and redirecting them will cause you to move and control your movements. Swimmers push water behind them in order to propel themselves forward. In the vacuum of space, there is nothing for you to push against, so moving your arms won't do anything.
We can have one key takeaway here, however. When you push on a fluid, you are redirecting it away from you, which causes you to move in the opposite direction of where you pushed it. Are you seeing a connection? It doesn't have to be a fluid!
We can assume the total momentum of the combination of astronaut and toolbox is zero, and it will remain zero no matter what the astronaut does to the toolbox because there are no external forces in the vacuum of space. Therefore, she can simply throw the toolbox away from the spacecraft to propel herself towards the spacecraft.
The reason for this is simple. The total momentum of the system must remain zero, so her giving the toolbox some momentum away from the spaceship must mean that she gains an equal but oppositely directed momentum (technically due to Newton's Third Law) towards the spaceship, which translates to velocity directed back towards the ship!
How fast she moves, however, depends on how hard she can throw the tools (how much speed she can impart to them), because the momentum she gives to the toolbox is the same as the momentum that she gains.
Now, this lesson may not have been the longest, but its results will be the focal point of a lot of the later lessons in this unit. Remember the results we have derived here, because they will be useful later on! With that said, we can move on to talking about center of mass, which is also a key concept that leads itself well into our discussion of more specific momentum topics later on. Enough said, hit that button to move on!
The AntiMatter Lab 