Business and life lessons from a freshman physics problem

Originally published on Wed, 05/27/2015 - 14:47

Some schools offer an introductory physics course that could be dubbed "physics for poets."   MIT offers "physics for masochists" – 8.012 / 8.022 – which, being a glutton for punishment, appealed to my loving a challenge.

I remember a particular problem which was simpler than most in the problem sets and required no math beyond simple algebra but was very rich in physics.  Now, some years later I see the problem as illustrative of the challenges of business and life.

Here is the problem:

We have two SuperBalls (extremely elastic toy balls made by Wham-O). The masses are $m_1$ and $m_2$ where $m_1 \gg m_2$.  The balls are initially stationary being held vertically touching one another at height $h$ above a flat horizontal hard surface, the less massive ball on top.  The diameters of both balls are small compared to $h$.  Both balls are simultaneously released and allowed to free fall.  How high will the less massive ball bounce?

That's it.  I've actually tried it with SuperBalls of different masses.

The point of the exercise is to get think about:

  1. Problem setting. In this case, the configuration of the SuperBalls initially and as collisions occur.
  2. Constraints. The physical laws that apply, here, conservation of energy and momentum.
  3. Simplifying assumptions. The things that we can ignore.  In this problem, as in life, the judgment calls that we make in our thinking processes are critical to success. The assumptions here are:
    • $m_1 \gg m_2$.  The algebraic solution hinges on this at several points (for example, $m_1 + m_2 \approx m_1$).
    • The diameters are small compared to $h$.
    • The use of SuperBalls and a hard surface mean that any collisions are elastic.
    • We are in a uniform gravitational field (constant $g$).
    • Trajectories are one-dimensional (we drop carefully or have some kind of frictionless guide mechanism).
  4. Immediate implications. The answer to the problem.
  5. Lessons to be drawn. More general insights that can guide our thinking and behavior in the future.  (What might we think about both similar and different situations?  What analogies might be drawn?)

Now I won't bore you, dear reader, on the details of the solution.  (The less massive ball bounces to height $4*h$, by the way.)  But there are a few comments that I wish to make about the problem and beyond.

  1. Problem setting. I think sometimes we all have trouble breaking actions into their constituent parts. The "five whys" technique of root cause analysis is motivated here.  In physics, business, and life the most apparent or obvious actions or events usually have deeper counterparts that must be understood. The insight that you need to solve this problem is that there is a sequence of actions from the initial drop to the small ball reaching its maximum height. Think of it the sequence as this:
    • Initially both balls have 0 velocity but are accelerating downward at 9.8 m/s/s.
    • The massive ball contacts the ground.
    • The massive ball reverses direction – and is now headed back up. The less massive ball contacts but it is still headed down.
    • Both balls are headed back up after colliding with each other. People who struggle with this problem usually don't see that the solution derives from the conservation of momentum and energy at this collision.
  2. Constraints. Every situation is governed by constraints, symmetries, boundaries and factors that control the flow and conservation of material and energy.  The clearer grasp we have of the principles at play and the forces that we need to grapple with, the more effective we can be in using these constraints to our advantage instead of detriment.
  3. Simplifying assumptions. Again, these are the things that we can or should ignore.  When I have read business cases (such as those in the Harvard Business Review) it seems to me that much of the problem is in how to separate out the critical elements from those which are to be ignored or left alone or delayed for later consideration.  So it is with most human interactions.
  4. Immediate implications. The reason problem sets are given to students is to develop skills for new problems and to develop the imagination to ask new questions.  But once in a while a particular problem and solution takes on a meaning of its own.   Dropping two SuperBalls and getting one to bounce four times higher -  well, that's just plain cool!
  5. Lessons to be drawn. How often in life are we just a small SuperBall – able to bounce and ready to make our mark in the world.  If dropped on our own, we can bounce back as high as we started.  But when we can bounce with and off of other SuperBalls that are bigger, smarter, wiser, or otherwise better endowed, sometimes we can bounce to heights we could not previously imagine.

We all can play the role of the massive SuperBall sometimes and the smaller SuperBall other times.  Teams can exhibit superlative performance when everyone at times is the massive SuperBall and at other times is the smaller SuperBall.