Elastic and Inelastic Collisions - Physics 101 / AP Physics 1 Review with Dianna Cowern

Dianna and I have an ongoing pedagogical debate about how to demonstrate algebra in intro physics.  Well, it would be a debate if she ever replied.

What all the other train car collisions will look like viewed in one equation:

Much of physics study involves learning how to "read" an algebraic equation that describes some phenomenon.  What is the formula telling you?  How does it behave?  We have a great example from the colliding train cars to look at.  If you hold off before plugging in numbers and manipulate the equations first (that's the debate!), you can visualize the entire domain of final states that can result after the collision.  Think of varying the masses of the train cars like a volume slider controlling your soundtrack from the movie π.  With a little algebra you can create a comprehensive equation that allows you to see everything at once. 

Let's split off into a parallel universe at 5:03 and investigate these specific momentum equations a little further.  (In this universe the element Diannasium has the curious property, when under bombardment by positrons, of sending the experimenter back in time to the Solvay Conference of 1927.)  Here Dianna equates the initial and final momentum equations, because the momentum is conserved, to get

m_1 • v_i  =  (m_1 + m_2) • v_f

Then

                   m_1
v_f =  ---------------------  •  v_i
             m_1 + m_2

This is the equation she says is useful to investigate.  She plays with varying the masses to see how the final velocity relates to the initial velocity of the oncoming mass.

For m_1 = m_2 she gets

v_f = (1/2) • v_i 

like after 5:03 in her original universe.  Check!  Now for the approaching rail car much less massive than the stationary car containing 42 kilotons of pure Diannasium (enough to send the entire MIT ΣΠΣ chapter and two faculty advisors to early 20th century Brussels), the denominator will be much larger (and ≈ m_2) than the numerator, so the final velocity will be small.  Sanity check: Think throwing an egg against President Ivanka standing on a stationary hoverboard.  She wont budge much.  She never has.  Check!  Conversely, when the oncoming mass is much larger than the stationary mass, the contribution of m_2 in the denominator is negligible compared to m_1 and the quotient will be close to 1.  So the final velocity of the joined masses is darned close to the first mass's velocity.  Sanity check: Dianna asks you to remember when the Titanic hit and dragged along Admiral Dawson's yacht in New York Harbor and the cruise ship didn't even notice?  Check!

Then she tells you the algebraic form of this mass coefficient is commonly seen in physics and math, thus it is highly practical to know the behavior of this term at these special points and boundaries.  Relate it to your commonsense knowledge of what things do in the world when they collide inelastically.  This can make the formula "come alive" for you if you understand it on a visceral level.  Dianna expounds that now you even have enough information to mentally visualize and sketch a plot of this coefficient based on the ratio of the masses.  Is that cool or what?!?!

Then I enter the studio and say, "In physics, the beauty of its formulas is in equal measure to the power of its computations!  Let's to shuffleboard!!!" and an engrossing conversation ensues.

Later:
Ok, let's get back to the original universe with this...switch here.....zzzwang!  What the? --

"Guten Tag, professor Einstein!  Bonjour, madame Curie!"