Keep the cogs turning!

The phenomenon of “learning loss” during the long summer holidays is well known so to keep your physics cogs turning, here’s a brief challenge.

I took my camera to the Five Mile Road on Saturday and photographed cars going past. The camera was set to 5fps so every sixth picture should have been recorded exactly one second later than the first picture, remembering that you need two events to define one time gap. The composite below shows a sequence of six pictures taken as a Mazda MX-5 drove past.

I also took a tape measure to record a scale for the pictures, having previously noticed that there are fence posts along the side the road. The mean distance between the posts was 3.2 m.

Your task is to estimate the speed of one particular car as it went past. You can download the full resolution picture if you want to be as accurate as possible in your calculations.

There are six post-gaps between the red markers, which equates to an actual distance of 19.2 m, and we know that the time interval is one second. Speed is the change in distance with respect to time, or simply distance divided by time. Therefore, the speed is 19.2 m/s (19.2 m of distance divided by 1.0 s of time). If you’re happy to accept this result and get back to enjoying your summer then that’s fine but it would be useful to check whether that answer is likely to be correct.

Jersey’s speed limit is 40 mph, which is 17.88 m/s (based on 1609 m in one mile). This puts our calculated value in the same ball-park and suggests that our method is likely to be along the correct lines.

If you want to refine this answer, then the first thing you’ll probably have spotted is that different parts of the car align with the posts, so 19.2 m is not the true distance the car travelled.

We can allow for this by measuring the distance using the same parts of the car (it’s front) and comparing this with the scale distance. If you are doing this yourself then remember to print the full resolution picture so you can measure the distance accurately – but I will use on-screen measurements for a quick answer.

The distance between the red markers is 22.2 cm on my screen and the distance between the blue markers is 21.7 cm, meaning that the car actually moved 21.7/22.2 of the 19.2 m distance. This is equal to 18.8 m, which is getting closer to the speed limit that should be observed.

The distance between the posts was a rounded mean figure from a set of readings that were; 3.15, 3.20 3.15, 3.17 and 3.18 (one of the posts didn’t have a convenient split into which the end of the tape measure could be secured). Using the mean with all three significant-figures, which I’ll leave you to work out for yourself, gives a total distance of 19.02 m for the red markers, bringing the speed down slightly further to 18.6 m/s, based on a distance of (21.7/22.2) x 19.02 being covered in one second.

Many of you will be happy to settle for the fact that the speed of the car was indeed 18.6 m/s but we can improve on the accuracy of this result further. This is going to mean moving beyond the GCSE Physics course but the thinking is still fairly accessible…

The next thing to consider is the camera’s frame-rate, which could be wrong. I checked this by photographing an old-fashioned stopwatch as its needle went around. A full rotation corresponds to 10 s and there were 50 pictures from start to finish (giving 49 time intervals). That means the camera was recording pictures at 4.9 frames-per-second. Of course, this assumes the stopwatch was correct but we have to take something as a starting point (much like the standards used for SI units).

A rate of 4.9 fps means each picture is taken 0.204 s after the one before and that means five intervals equates to 1.02 s (rather than one second, which we assumed previously). This in turn drops the speed by a factor equal to 1/1.02, giving 18.2 m/s.

Finally, we need to consider the problem of parallax, which is caused by the car having been slightly in front of the fence posts, so it actually moved a shorter distance than we think. (This is the same effect as holding your hand out and framing the moon between your fingers: the moon is clearly a lot bigger than the space you create but it’s further away so can be seen within the smaller gap.)

The easiest way around this problem is to research the true length of a Mazda MX-5 and to check how many car lengths the vehicle moves in one second. (You may recall having used this approach to find the speed of some aircraft in Jersey’s annual air display back in the autumn term.)

If you’ve read this far then you’re pretty keen so I’ll leave you to carry on from here.

(HINT: The real-life car is almost exactly 400 cm long and remember that the time period is actually 1.02 s for the two-car picture shown above. I get a final answer of almost exactly 40 mph but that is still subject to at least one further error that you might be able to spot if you look very carefully at the two images of the car. Consider whether the car was travelling exactly perpendicularly across the field of view and how that would affect measurements of the car’s length.)

In the final analysis, you are likely to conclude that the car was travelling at 40 mph, which is reassuring for the driver given Jersey’s speed limit!

If anybody (in my current physics classes) wants to send me his/her own calculations and final answer then glory, my personal admiration and possibly some sort of reward all await!