How To Use Conservation Laws

42 How To Use Conservation Laws

Last chapter, we learned that momentum is conserved in the absence of external forces, which we summarized using the relation

$\vec p_{\rm f} = \vec p_{\rm i} + \vec J_{\rm ext}.$

This is our first example of a conservation law. Conservation laws have two critical features that make them extremely useful:

Conservation laws relate the beginning of a story to the end of a story.
As long as the assumptions of the conservation laws are valid, the details of what’s happening in the problem are completely irrelevant.

These features are so powerful, I like to refer to conservation laws as “physics cheat codes” because they can turn really hard problems into really easy problems. Let’s see how this all works.

Exercise 42.1: Collisions

Curling is an olympic sport in which teams take turns trying to slide a stone on ice towards a target. The closer a stone gets to the target, the more points you get. The teams take turns sliding the stones towards the target, and you can use your own stone to try to hit the other team’s stone out of the way. Here is a pretty amazing example of this.

Let’s consider the collision of two curling stones. Stone B has a mass of 17 kg and starts at rest. Stone A has a mass of 19 kg, and is slid towards stone B. It hits stone B at a speed of 0.2 m/s. After the collision, stone A is moving at a speed of 0.05 m/s at an angle $\theta=20^\circ$ relative to the incoming direction. Determine the outgoing velocity of stone B.

This is our first momentum problem, I will try to walk you through it. Try it! If you find yourself unable to figure out, I go through the full solution in a video at the end.

Step 0- What’s the story? We now have two things we know about physics at our disposal: Newton’s laws and conservation of momentum. Consequently, from this point on you will have to figure out if the problem at hand is solved with forces or with momentum. So- what’s the story?

The problem tells us the state of a system (the two stones) before the collision, and then asks us to say something about the system after the collision. Moreover, you don’t really care about what’s happening in between. Thus, this is “before and after” problem, which leads us to our rule:

“Before and after” problems are solved using conservation laws.

Moreover, in this problem we have two colliding stones, which leads us to our second rule:

Every collision problem is solved using conservation of momentum.

The story starts right before stone A hits stone B, and ends right after stone A hits stone B.

Step 1- Draw a picture. We can define the x-axis so that stone A comes in along this axis. Further, let’s define the. y-axis so that the outgoing velocity of stone A is upwards. Label everything. Moreover,

The picture for a “before and after” problem must include a before and an after, both clearly labelled.

Important: You should use different letters for the incoming and outgoing velocities. Let $v_{\rm A}$ and $v_{\rm B}$ be the incoming velocities, and $u_{\rm A}$ and $u_{\rm B}$ be the outgoing velocities. This will help you keep track of what the symbols mean when setting up your equations.

Step 2- Find relations. We already figured out that this is a momentum problem. Moreover, since the collision happens on ice, we can approximate the collision as happening on a frictionless surface. Therefore, there are no external horizontal forces, and momentum is conserved. Go ahead and setup $\vec p_{\rm f}=\vec p_{\rm i}$ for the system.

Hint: remember momentum is a vector, so the conservation of momentum equation is really one equation for each axis.

Step 3- Solve unknowns. Do the algebra to find $u_B$ and $\phi$ .

Because this is our first momentum problem, I decided to include a video with the full solution in case the bits that I wrote above are not quite doing it for you.

Before moving on, I want you to appreciate what just happened. The forces involved in the collision are incredibly complicated! The stones hit each other, and slightly deform each other, and then bounce back. The forces are most certainly not constant, nor easy to describe. We don’t know how much the collision lasted either. Fortunately, we simply don’t care. All those details are irrelevant: what happens during the story just doesn’t matter. Our conservation law allows us to relate the final state of the system to the initial state, regardless of the details. That is why conservation laws are so useful!

To summarize:

“Before and after” problems are most easily solved using conservation laws.
The picture for a “before and after” problem should include a before and an after.
Every collision problem is a conservation of momentum problem.

Let’s drive these points home with more problems.

Exercise 42.2: Nuclear Decay

You probably know that atoms have a very compact nucleus made out of protons and neutrons. A plutonium 239 nucleus (meaning the number of protons and neutrons is 239) can spontaneously decay by emitting an $\alpha$ particle comprised of 2 protons and 2 neutrons, leaving behind a Uranium 235 nucleus. The nuclear reaction looks like this

${\rm Pu}^{239}_{94} \longrightarrow {\rm U}^{235}_{92} + \alpha.$

The $\alpha$ particle is ejected with a velocity $v \approx 1.7\times 10^7\ {\rm m/s}$ . Determine the recoil velocity of the Uranium nucleus.

To make the math simple, you can assume that protons and neutrons have the same mass (this is approximately true), which means you can measure mass in units of “number of nucleons.” So, for instance, the mass of ${\rm Pu}^{239}$ is $m_{\rm Pu}=239$ , and the units are “number of nucleons.”

Now draw a picture. Make sure you show the before and after!

Notice this time I used $u$ and $v$ for the velocities of the uranium and $\alpha$ particles. This made more sense in this problem, since the initial velocity is zero for everything. The point here is to use your notation to help you keep track of variables. While we could write $v_{\rm U}$ and $v_\alpha$ , I find it much easier to distinguish variables if they have different names in the first place.

I hope you are dully impressed: nuclear decay is an inherently quantum mechanical process that is difficult to describe in detail. Nevertheless, we were able to solve the above problem without knowing anything about nuclear physics!

Let’s do one more.

Exercise 42.3: Baseball

A baseball has a mass of 145 g. A pitcher throws a 98 mph fast ball, but the batter connects: the baseball flies off at a $35^\circ$ upwards angle and a speed of 110 mph. The collision between the bat and the baseball lasted 2 ms.

A. Estimate the average force that the ball experiences during that time.

B. In a typical major league baseball field, the outfield is $\approx 400\ {\rm ft}$ from base. Will the above hit be a home run?

The above solution was surprising to me at first. We calculated that the height of the ball as it crosses the outfield is $\approx 40\ {\rm m}$ . This is absurdly high: for instance, a 10-story building is $\approx 30\ {\rm m}$ tall. Clearly, something about solution is incorrect. Can you figure out what? Hint: it’s not the math that’s wrong!

Key Takeaways

42 How To Use Conservation Laws

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