How Do You Measure That (HDYMT)?

Consider the following question:

A projectile leaves a canon at 25 m/s. If the barrel of the canon is 1.0 m long, a) what is the average acceleration of the projectile? b) If the projectile has a mass of 200g, what is the average force on the projectile while it is in the barrel of the canon?

That is a fairly typical physics problem, but as we have already discussed, it suffers from a surplus of explicit information, transforming it from a problem to a calculation. Plug’n’chug rather than critical thinking.

I prefer to present the problem this way: bring out a canon, shoot it, and ask students to determine the average force on the projectile in the barrel. In my case, I have an air canon that we built years ago that still serves me well. It is made of ABS pipe, and can shoot a squash ball or ping-pong ball. Following the GAP method of problem solving, it is a fairly straightforward process to work backwards to solve the problem:

  • We need force F
  • F = ma
  • m is measurable with a balance.
  • Solving for a requires 3 variables. 1) Initial speed in the canon is 0; 2) Length of the barrel can be measured with a metre stick; 3) Final speed is the speed of the ball as it leaves the canon.

So the entire problem can be solved once we know the speed of the ball. But how do you measure that?

At lower speeds, video analysis works reasonably well. But as the speed increases, the video frames become increasingly blurred and harder to measure accurately. And who wants to stay at low speeds? After some experimentation, we eventually settled on using a strobe light at 100 flashes/s recorded on video. We then extracted the few frames that captured the event – each with 3-4 strobed images – and stacked them in photoshop. Here is a sample result:

With a metre stick in the frame, we could determine the distance between images of the ball, each of which were .01 s apart. Measurements could be made manually, or with the assistance of Tracker. Manually, students can use a ruler to measure the distance between ball images and compare that to the metre stick scale – and this can be done on paper, or even on a computer screen. Using tracker, the scaling can be done automatically once the metre scale is set (Tracker will set 10 frames to a single still image – for more frames the same image can be loaded multiple times, 1 for each frame needed).

Once the launch speed is determined (and this itself is a touch tricky – the ping pong ball experiences quite a bit of air drag), we just follow the steps of the gap method back to the answer.

Interestingly, the first time I gave my students a still image problem like this with a metre stick in the image for scale, they were completely baffled. Even though scale images is something they have done in both Geography and Math, they seemed at a loss applying those skills in Physics. They kept asking for the formula to scale the image. That tells me I need to do a lot more of this sort of thing!

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