Writing about Covid-19 in essay 305 a question arose to explore here. I need no trigonometry, just Y8/9 memory of set square ratios and the simplest Pythagorean triangle.
A person occupies a so-called bubble of space and this may be occasionally trespassed by someone going past. Let us assume that the relative movement is such that we can treat one person as stationary and the other as moving in a straight line.
Here's a general diagram to describe this situation. The person being passed, Olive, is obviously at O. The person going past travels along line AB, closest point C.
1. Let's start with r=2 and d=1. Find the length AB. Let us assume all lengths to be in metres.
2. What's wrong with this is Olive has no size. Social distancing is some measure from the outside of Olive to the outside of the next person. So let's imagine that Olive's personal space adds half a metre. So r=2.5. Leave d=1 and again calculate AB.
Let us try to imagine a more general case, but continue to perceive AB as defining the closest approach of the nearest edge of the passer. So if this is a cyclist or a runner, this is the path of the nearer elbow. d is the length cut off Olive's bubble and OC is not the residual bubble size. Here's a slightly better diagram.
Let's worry about this situation from Olive's viewpoint. She wants a bubble of two metres, so r is around 2.5, maybe a little smaller. Her perception is that anyone within that bubble is threatening her with infection, but if the trespass is brief enough, she will reluctantly tolerate this. So suppose her next margin of upset occurs when the closest approach is 1.5 metres, which makes d=0.5.
3. Find length AB when d=0.5, r=2.5
Olive doesn't care about the length of AB. She cares how close someone gets when they go past and, when she thinks about it, how long that person is inside her safety bubble. Which is exactly why we want to know the length AB, of course.
Suppose that the difference in speeds is 3kph, a slow walk.
4. How do you convert 3 kph into m/s?
5. How long does it take to travel line AB at 3kph?
Olive feels that a walker going past her really calls for her to have managed to do some avoidance and that ideally both parties would have taken action. If she is being overtaken, so that the other person is going twice her speed, then there comes a point at which she will not be aware of being passed until perhaps too late to respond. But if Olive is only doing 2kph then a 3kph difference puts the other walker overtaking at 5kph and again she thinks this is slow enough she ought to hear and respond. Olive doesn't often walk that slowly, so her first concern is over cyclists, who tend to be going relatively quickly and change direction with difficulty. She thinks about this and decides that a fast bike has less control, so should she worry if her bubble is squeezed? A cyclist at 1.5 metres is about as close as she would stand if there were no worries about infection, only about collision. Let's experiment:
6. If the cyclist is going 10kph faster than Olive, how long does it take to travel the AB in Q3? What about twice that?
So, if a slow cyclist is a problem for about a second, is that a problem of infection? Olive thinks that the issue of infection is to do with the other person breathing out, so her action on being passed from behind is going to be to turn away so as not to breathe expelled air from the cyclist. If she is passed form in front the relative speed is much higher and she would normally expect that not to be any sort of surprise, so she would equally expect to have moved out of the way so that her bubble is protected.
Which then leaves her with a problem about joggers and runners. From her perspective these come in two types, those she can hear and those she can't. Again it is those from behind that are an issue; the loud ones, in general, are giving warning and she can move to one side so as to leave space. Or, indeed continue exactly as before to leave the runner to make a decision about how to avoid trespass or collision.
7. If a runner is doing 8-10 mph, how long would they be in the bubble if the closest they reached was the same 1.5 metres of Q3? This is a harder question, because you have another intermediate stage.
8. Sometimes the path narrows and she is left with no choice but to accept a brief moment at just one metre as in Q2. Repeat Q7 with this change.
Olive would really prefer for the faster mover to take responsibility for their actions and proximity. The more she thinks about this, the more sense it makes that choke points, those places where proximity might be forced if attempting to overtake, were places that cyclists and runners changed speed to match everyone else.
Olive used to ride a bike; she remembers there being no problem with straightforward walkers. The unpredictable walkers had appendages, like small children or dogs. Or worse, both children and dogs. Those people find it very difficult to see any further than the bubble that surrounds their party and also act as if quite surprisingly deaf. Olive wonders how often a cyclist or runner is brought down by dogs. Q9; research this.
DJS 20200616
1. OC is clearly also 1 unit of length, so ∆OAC has sides 1 : √3 : 2, c is √3 and AB is 2√3.
2. d=1, r=2.5 so OC=1.5. So ∆OAC is right angled, hypotenuse r=2.5, OC=1.5. This is a 3-4-5 triangle, c=2, AB=4.
3. I see this as another 3-4-5 triangle. ∆OAC is right angled, hypotenuse r=2.5, OC= 2. This is a 3-4-5 triangle, c=1.5, AB=3.
4. 3 kilometres per hour is 3000 metres in 3600 seconds, so 3kph = 3000/3600 = 30/36 = 5/6 = 0.833 m/s.
5 3 metres covered at 0.833 m/s is 3 / (⅚) = 18/5 = 3.6 seconds
6 Relative speed 10kph = 10/3.6 m/s = 2.78 m/s, so the 3 metres of AB takes 1.08 seconds, A 20 kph difference would take 0.54 seconds.
7. 1 mile = 1609 metres, call that 1.6 km. 10 mph = 16 kph = 4.44m/s. 8 mph = 12.8 kph = 3.55 m/s.
So AB=3m suggests conflicted times of 0.675 and 0.85 seconds.
8 AB = 4 instead of 3, so the times lengthen, to 0.9 and 1.125 secs.
9 Runners colliding with cyclists requires both to be on the same route. Runners on such routes every day would collide once or twice a year, probably only where one or both are concentrating on speed not route. See Essay 305. Dogs and cyclists in proximity is a disaster; one or the other is going to get really hurt. Runners and dogs is more common and a daily run on routes with dogs close by will produce tripping dogs about every third week. One learns to beware, but the random movement of a dog you have not seen ahead is a real problem. I was tripped up today by a dog-walker letting his two dogs out of his car with no leads. He was quite oblivious to my approach, as predicted (and I was loud at the time, last 100m of a fast run for the day) but I couldn't tell that this was a man with a dog ! I was even louder at the dog (who at one point was within an inch of pounding feet) even having tried to avoid the dog while it was well inside my reaction distance. A raised finger moment, but which?