## Reverse Engineering

…[O]ur barrier does not move so when our car carrying the surfboard hits it, it’ll come to a complete stop, and all that momentum from the car will be transferred to the surfboard.
Tori Bellacci

I want to give the Mythbusters some props for trying to introduce the concept of momentum; however I must instead scold them for completely bungling it.  In the scenario they are testing, they want to see how lethal a surfboard is when flying from the roof of a car.

I’m not really sure what they think momentum is, but let me clarify.  An object’s momentum, $\vec{p}$ is simply its mass times its velocity, $m\vec{v}$.  It is probably my favorite physics quantity because it is always conserved.  That’s a very useful foundation for solving lots of problems.  Anyway, does the car transfer any (or all) or its momentum to the surfboard?

Well, before it hits the barrier, the surfboard has x-momentum equal to its mass (let’s say 10 kg) and velocity (40 mph = 18 m/s):

$p_{1}=mv_{1}=(10 kg)(18 m/s)=180 Ns$

I don’t think it’s unreasonable to assume that there was minimal friction between the car roof and the surfboard, such that it left the car going 40 mph.  Since its mass didn’t change, its momentum after the collision is (surprise, surprise):

$p_{2}=mv_{2}=(10 kg)(18 m/s)=180 Ns$

The car doesn’t transfer any of its momentum to the surfboard.  The surfboard simply retains its original momentum, which makes sense when you consider a less frequently cited definition of force as the time rate of change of momentum:

$\vec{F}=\frac{d\vec{p}}{dt}$

There are no forces acting on the surfboard during the collision (though there are LOTS of forces acting on the car), so it’s momentum is unchanged.  The only reason you’d want the car to be stopped pretty fast is that there is a little friction between the surfboard and the car roof, and you don’t want it to last very long.  Think about pulling a table cloth out from under some dishes.  If you do it fast enough, it has a minimal effect on the dishes (surfboard).

When they take their board to NASA for the water tunnel experiments, we have the opportunity to think about some more sophisticated concepts.  For something passive like a paper airplane (or a surfboard) to fly “stably”, any perturbations from its preferred orientation must induce forces that will correct itself and restore the original orientation.  That is, as the surfboard began to pitch down, there would be a pitching moment created that would prevent it from nose-diving into the concrete.

If you ever played with paper airplanes, maybe you used paper clips to change the location of the center of mass so that your paper airplane was more stable.  Without the center of mass in the correct place, your poorly designed paper airplane would veer off course.  The surfboard in this case is like a poorly designed paper plane.  The team observed with Kari’s surfboard that it would pivot wildly in NASA’s water tunnel.  A stable surfboard would have maintained its orientation to the flow even if it were nudged up or down.

I’m not really sure what they try to accomplish with the 3D-printed surfboard.  Instead of letting it pivot, they clamped it at the back so that it wouldn’t move… which kind of defeats the purpose.  Of course it’s a streamlined body, so it’s going to have a quiet signature, but I don’t see how that is informative when it come to the matter of stable flight.  I’ll cut Kurt Long some slack since I’m sure the many hour conversation was edited quite heavily.