Speed is distance divided by time. Average speed is total distance divided by total time.
1 mile = 1609.344 metres, 1 hour = 3600 seconds, 1 gallon = 4.54 litres - all ‘known’ information.
This page follows on from Essays 156-162 on vehicle developments. A principle being applied is that results are interpreted in the context of the question. It is of no use to society if your ‘answer’ is merely a number; it needs to be a response in terms that are understood, it needs to be presented in an intelligible form It needs also to recognise the approximation level - if input was all at one s.f. then the answer cannot be more precise; therefore a good answer recognises the range of input values and supplies a range of output values. A better answer recognises which inputs could be better measured, identifies critical inputs and may even qualify outputs based on further assumptions taken from context. Yes, you have to answer the question in exams, but answering the question outside that narrow field is more like answering all the implied questions, too. The learned exam technique is to answer what was posed and you must still do that; in the world of work you should be prepared to go far further - “This is what you asked about, but this may be what you wanted to know”.
A topic early in the Y12 maths course looks afresh at distance-time and velocity -time graphs; the latter are great because the area under the graph represents distance and the subject is a nice easy beginning to Sixth form work as the content was introduced maybe as far back as Y9, so all that has changed is the expectation of behaviour (volume of work done, enthusiasm, willingness to go further, etc). Here are some classical questions, using g=10 m/s/s.
1 A train company is looking to add a stop on the main line. The expected speed of the train is 144kph; braking and acceleration are around 0.2g. The stop is to be one minute. (a) How much time will this add to the timetable? (b) What braking / acceleration would add only five minutes?
(c) Make comment about adding or removing stops to a Newcastle to London rail service, bearing in mind that you can sell a three hour trip but you’d like to sell a two hour trip from further north than York (as now).
2 Driving on the motorway at 70mph, you intend to stop to refuel. The stoppage time is 5 minutes, the extra distance travelled is 400m at 20 mph and use g/3 for both braking and acceleration. What is the total extra time for such a stop?
What figures do you think F1 teams work on? A tyre stop takes 2-4 seconds but that is not the extra time to add to a lap, since there is extra distance, slower speeds and accel/decel to account for. Can you work up some figures and assumptions to support this data: at the Hungarian GP of 2014 the fastest 35 pit stop times were from 21.6 to 24.2 seconds. Drivers had from one to three stops (e.g. at 3 stops were: Rosberg, 2nd; Massa, 5th; Bottas, 11th).
3 If you only fill your tank half-full, consumption while the tank empties improves by 10% over tank-to-empty consumption, 20 km/litre (k/l). If on a long enough drive to add just one stop by this behaviour, what is the cost or gain? Take an assumption of range as 1000km and find (i) the tank capacity (ii) half-tank consumption and range. Assume ‘empty’ leaves enough in the tank to not have issues and so find (iii) the extremes of trip distance that demand an additional stop (between fuel stations). By looking at the fuel volume consumed, express a percentage difference between the two strategies and make some comments about other costs.
4 A group of walkers claims to cover the ground at 5kph but over a day achieved more like 4kph. If you assume they were ‘out’ for eight hours, how much time did they spend stopped? If their planning included an hour stopped but still assumed 5kph over the ground and if the result of the day still said they’d done 4kph, how much extra stoppage time was taken?
A different group estimates they’ll walk at 5kph but stop for 10 minutes an hour; what then is their distance each hour? How fast would they need to walk to produce an average of 5kph?
5 A group of walkers realises that they walk faster on any sort of sheep track than they do over heather or long grass. If the two speeds are 5 and 6 kph, then
(i) How far off course (degrees off direction) would they be prepared to accept for a material gain in covering the ground?
(ii) How does this change for lower speeds, say 3 and 4?
(iii) Suppose the difference of speeds was only 0.5 kph; find what sort of difference of direction will be worthwhile. So you could use algebra, you could use a range of values and draw a table - it is up to you how you do such exploration.
(iv) Make comment about how these results affect your walking strategies.
6 In orienteering and mountain walking there is a rule named after William Naismith. This rule distinguishes between ground speed and additional time to climb hills. A typical set of figures would be 4 kph and 600m per hour. Some people would do corrections for downhill and terrain, but we will ignore those for now. Assume all ground under discussion to be equivalent (think of grassy hillsides cropped by sheep). People who decide to walk further at a shallower angle of ascent (zig-zagging upwards) are treated as walking directly uphill for the speed component and the extra time is part of their ascent rate.
(i) How long will it take to climb Helvellyn with no stops included, on a route requiring 12km and 1000m climb?
(ii) I’d like to climb Fairfield too on the way up (it’s adjacent and a little smaller) and I can see a route that adds only 1km but there’s a descent of 400m so I assume that means extra climb. How long will this take?
(iii) This route takes me around Seat Sandal. I wonder if I want to go over it or around it. The map says the extra climb is only 150m so how much shorter would such a route have to be to make no difference?
(iv) In Jan 2015 on a cold day with ice above 600m I did Fairfield, starting at Rydal after 10:15 and finishing at Dunmail Raise at 13:15. I see that as a little over 10km and 800m climb. I think I lost time, around 30 mins, on a steep snowfield I crossed twice. Is this consistent with my expected 5kph and 1600m/hr?
(v) I then went up Wythburn, over High Raise and down past Stickle Tarn to Langdale, in a bit under a further four hours. I think that is another 11km and at least another 500m climb. Wythburn was horribly wet and the descent to Langdale slippery. If we assume I went no slower (it might well have been faster, because it wasn’t so cold), how much time did I lose in coping with the terrain?
(vi) You look up these mountains and find routes that fit (iv). Start and finish on the A591. Correct my data from your research. Showing your calculations, indicate how long you think this walk would take you. Particularly, decide if you’d (ever) go over Seat Sandal, showing why.
(vii) What would you have done on foot at 13:15 at Dunmail Raise with sunset known to be at 16:30 (so it’s dark at 17:00)? I was in the shower in my room in the New Dungeon Ghyll Hotel by 17:15.
I am keen to encourage people into taking easy maths—there is nothing on this page outside the wit of a Year 8, except the trig implicit in Q5 (Yr 9)—and applying it intelligently to real-life problems. Part of the difficulty experienced at A-level is that the process of extracting information from a problem has not been practised. I think this is what extension work in Y9-11 should be, not taking work from future courses. Setting up a problem is a progressively bigger issue in maths, so we ought to learn how to do that far earlier than we do. I also think far too many people avoid using maths for trivial things (If I need eight hours sleep, it is after midnight, when will I be getting up? Since I have to be at work/school around eight, why didn’t I think of this sooner?) and my experience on Dartmoor simply emphasises this - bright kids failing to recognise as early as 09:00 that they have a problem meeting the expectations of the day, such as a finish at 19:00 (when doing arithmetic would tell them that 21:00 is now (already) more likely). It is not whether they can do the arithmetic, the issue is that they somehow refuse to do it; they don’t ask the question. Some of the blame for this attaches to that thinking that says we can’t do spelling in a maths lesson, can’t do chemistry in a physics lesson, can’t do numbers in a language lesson - putting types of thinking into subject-titled pigeon-holes simply spoils the ability to think at all. From the point of view of maths (bigger than arithmetic) or thinking (bigger than maths, I hope) this is a failure we could change within ourselves. Perhaps this is a life skill; sitting back from a problem enough to place it in context and ask whether the answer so far is useful - and, if not, what else needs to be done. Often, this produces the side issues characterised by, “Have I done enough?” and “What will it cost to take this further?” but hopefully, gets us all past the truly terrible “Does that get all the marks?”
DJS 20150324
1 144 kph = 40 m/s. v/a=t T=t1+t2+t3 = 40/0.2 + 60 + 40/0.2 = 860 secs = 14:33 mins
Using 0.4g gives 4:20. Using 5 minutes, leaves 240 secs for speed change, so if equal, g/3.
Newcastle service generally stops at Durham, York, Grantham, Peterborough, Stevenage and King’s Cross, taking 3 hours. removing five stops might save 30 minutes and offer a sub 2:30 service. How about once a day, Inverness, Edinburgh, Newcastle, Peterborough, King’s Cross? Just who does the job of deciding this and do you think they have the wit to go ask the right questions? Would you? Is your answer different now to 20 minutes ago?
2 20 mph = 8.94 m/s, 70mph=21.3m/s. speed change time is 104 secs, plus 44.7 at slow speed plus 5 mins stopped => 7:29 total.
3 The stop costs around ten minutes (Q2) gain. A tank is 50 litres. half-tank consumption is 22k/l, so 25 litres =>550km range. So a ‘long-enough drive’ is, (iii) 550-1000km. So the half-tank consumption is 25 litres to 25 + 450/22 = 45.45 litres [10 gallons, surprisingly], while the full tank strategy uses 27.5 to 50 litres. These are both 10% (to 4sf, far greater than the data given). So the costs are to do with the stopping at all; since some stops will occur, adding some additional time at the pumps is the cost for 10% fuel improvement. However, the sort of person who would drive on a half-tank would not stop on the motorway (fuel too expensive) so the stoppage time would be significantly different between the strategies; a likely gain of 10p/litre, but an additional 20 minutes travel time.
4 8@4kph =>32km, but 32@5kph=>6.4 hours, so 1.6 [1:36] stopped.
7 hours expected walking =>28km, so 28@5=>5.6 hours => 1.4 stopped. In both cases there is around an hour and a half of extra stopping.
5/6 hour @ 5kph =>25/6 km per hour, 4.17 kph. To achieve 5 and stop for 10/60, requires 6 kph.
The issue here is that (too many) walkers are quite quick when actually walking, but stop altogether too easily. The difference between an hour at a mean 5kph and walking at 6kph is [5km covered in 5/6 hour, so] 10 minutes ‘lost’ on other activity. Across a day that is all too soon more than an hour. This drastically affects what you can do in a day. It is very hard for groups to walk for as much as 55mins per hour. I’ve done it with soldiers, but even TenTors 55 teams find it difficult to keep on task that much; yet this is what they need to do. I find I stop for 5 minutes in the first hour and, unless I have navigation issues, will then stop under two mins/hour. [Looking at the view, the map, exactly where I’m going; I usually stop to fetch something from my bag, but recently I’ve been trying to do that too while still moving.]
5 A right angle triangle with long sides 5 and 6 has an included angle of 33.56º, so a path at 30º off course gives virtually no advantage, while at figures up to 20º taking a sheep track roughly in the right direction is going to be worthwhile [21º off is still 12% faster than the direct route]. For speeds of 3 and 4 the critical angle is 41.4º, so 30º off course is going to produce appreciable gain [15% faster]. The bigger the proportional difference in speeds the bigger the acceptable ‘aiming off’.
Your use of this when out walking will depend on the state of the track and of the alternative. In the wet, and when the visibility is bad, you might prefer the direct route, but in sunshine the track will be faster. The two speeds (and their difference) will change with weather, too. This is a result worth applying, especially when you can see the distant target, such as in good visibility on Dartmoor. I used this in lessons (both in class and on the moor) at PMC, where a significant fraction of the school walked on Dartmoor often enough to listen.
6 12/4 + 1000/600 => 4:40:00, 4hrs 40 mins
13/4+1400/600 => 5:35:00 an additional 55 minutes.
150/600 => 15 minutes, so we need to be 1km shorter.
2015 January (snow, ice above 600m) I did only Fairfield in under 3 hours including a lost 30 minutes on a sloping ice field, a total of a mere 8km and 800m climb. I expect to do 5kph and 1600m/hr, so 8/5+800/1600 => 2.1 hours, so 24 mins lost on the snowfield not 30 - or around 6 minutes error in the figures. Yes, it is consistent, within the implied error bounds. Note any stopping (I do little of that) is built into the figures used.
- 1.(v)The second part of the day 11/5+500/1500 =>2:30 => 1:30 lost to terrain. Of this, a good 20 minutes was the steep descent into Great Langdale (yes, it is as fast to go up than down) as much again for the lack of light (sunset was when on the top of High Raise and I ran for a while) and that means I lost 50 minutes in that horrible boggy ground that is the whole of Wythburn, making that a 3kph valley, not a five.
- 2.(vi) I look forward to your figures. I expect most to conclude: (i) that they wouldn’t do this in the winter and if they did, to start very much earlier in the day (ii) that Seat Sandal is something to climb on a very nice day when full of vigour, or because you’ve done this before and know how bad the path around the bottom is (iii) that you probably stop often enough to bring your ground speed down to 4kph and that the old guy is still climbing quite quickly, which means that doing 20km in the winter is a tall order (iv) that in practice your map-reading is going to slow you down, as is stopping for each other and food, though good walkers will make those activities overlap.
- 3.(vii) Going by road would be more like 15km and 300m total climb, about 3:15, plus dodging the traffic - about the same, then. If I’d not made the mistakes on the snowfield I’d not have lost the light and so I’d have finished a whole hour earlier. No, I didn’t stop but for brief looks at the map and to add/shed clothes. Yes, I was hungry at the finish, but, believe me, dinner is good at the NDG. Total day 6:45 ; 21km, 2100m climb; at 5kph & 1600m/h with 55 mins lost to terrain. Conclusion: most people would deem me mad to walk back to the hotel and would have taken the bus (change in Ambleside, have lunch & tea too; Google maps says 1:34 plus waiting). Those who agree with my decisions please give yourself a congratulatory pat for being hill fit and confident in your abilities. Simon Evans [PMC, skier], for example. A quicker route would be to go down the A591 and then SSW to Easedale, past Easdale and Stickle Tarns, about 11km, far less climb and a better quality path (which I have done in the past), so probably 30-45 minutes quicker (or more relaxed walking; past experience told me that I dislike the mapwork and ground in crossing between the two tarns, which was why i took the other route. I won’t repeat Wythburn until it is dry.