Exploring distance time graphs
Exploring distance time graphs

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Exploring distance time graphs

1.9.3 Video task: graphing the journey

Now watch the video ‘Single-track minders’. There are four activities associated with the video sequence. Try the appropriate activity when you are asked to stop the tape. You should tackle Activities 25 and 26 at the end of the video.

Now watch the video.

Video, Click to watch part one

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Transcript: Single-track minders - Part 1

JUDY EKINS
his is the Severn Valley Railway, which runs alongside a stretch of the Severn River between Kidderminster and Bridgnorth in the West Midlands of England. There are only a few preserved steam railways like this operating in Britain, and all of them are privately run, relying heavily on the help of enthusiasts, who drive the trains, operate the signals and draw up the timetables. A timetable has to suit passenger demand, but it has another very important function: to ensure safety. This is especially important on single-track sections of line, like those on the Severn Valley Railway. If two trains collide, the consequences can be disastrous.
NEWSREADER (MICHAEL BUERK)
Five people have died in a train crash in Kent. Two trains collided in thick fog on a single-track line.
JUDY EKINS
Collisions on a single track are rare but, nevertheless, a very real danger. Even in the ’90s, trains have come to grief, despite what are thought to be fail-safe precautions.
NEWS REPORTER (VOICE-OVER)
The two trains collided head-on in thick fog while travelling on the single track line. It’s thought both were running late, and there’s evidence that one, at least, skidded as it tried to brake before the impact. The collision shattered the front diesel units. The two drivers are understood to be amongst the fatalities.
JUDY EKINS
In common with other railway lines, the Severn Valley Railway has to maintain strict safety rules. There are speed limits – here it’s ten miles per hour – and signals indicate whether or not it’s safe to proceed. There would be no problem in running just one train on one line, but as soon as there are two or more trains running at a time, they have to be able to pass, so you need to make use of locations, such as stations, where the line become double track. Bridgnorth and Kidderminster both lie on the valley of the River Severn, and the two are connected by 24 point five kilometres of railway line. The line starts on one side of the river, before crossing over a bridge to the other side. The line also goes through a tunnel outside Kidderminster. Trains are able to pass one another at a number of intermediate stations along the largely single-track line. A train not only waits at a station platform to allow passengers to get on and off; it may also have to wait for the line ahead to be clear. This train is having to wait for a train travelling from the other direction to arrive at the station, before it can proceed. Efficient timetabling should ensure that no train is held up unduly. In order to investigate this further, I went to Bridgnorth and boarded a train for Kidderminster with Keith Shaw, who has responsibility for timetabling the service. The first stop was to be at Hampton Loade, one of several stations and potential passing places on the way. Like all good railway supervisors, Keith is armed with a stopwatch.
KEITH SHAW
Ah, nine o’clock, right on time.
JUDY EKINS
Slowly but surely, the train gets up steam. But its departure time is being noted for a reason. Could you tell us what you’re doing today?
KEITH SHAW
Well, we’re just checking the times of the trains in order to compose a timetable – we’ve just left Bridgnorth now pretty well on time. We’ve had some difficulty starting, as you will have noticed, but that’s something that maybe we have to take into account in the timetable.
JUDY EKINS
You have problems with trains being late?
KEITH SHAW
No, not too bad. On the other hand, it is important that we don’t just sit back and look at the timetable and say, well, that’s it for ever.
JUDY EKINS
So, let’s look at some of the local landmarks and key places that affect the journey time. The train has reached a place called Sterns, just over half way to Hampton Loade, where there is a danger of the line slipping into the river, and so there’s a speed restriction, or ‘slack’, of five miles per hour.
KEITH SHAW
It’s 9.15, we’ve just passed the slack at Sterns, and as this is only a one-coach train, of course, it doesn’t take us long to pass it, but if you’ve got eight or nine coaches on, the whole of the train would have to pass the short section – I’ll just make a note of that.
JUDY EKINS
The particular train on which we’re travelling only needs to have one coach, because it’s the early morning taxi for railway personnel.
KEITH SHAW
Right, we’re approaching Hampton Loade now, and at Hampton Loade there’s a very interesting ferry, and there’s a ferry across the river which has no power, no motive power. It’s driven by the current of the river, and it’s two very elderly ladies operate it; if you want to cross you go and push a button on a pole, and they come out of their house and get in the boat, and it’s anchored to a wire which runs across the river, and by manipulating the rudder, they can use the current to drive the boat across and back. It’s quite fun. And occasionally when I see them at Hampton Loade, I use the ferry, park the car on the other side of the river and come across it – it’s quite nice. It’s quite a nice feature. Just approaching Hampton Loade station there, so we need to check the time. Just 9.20.
JUDY EKINS
So, that’s the journey as far as Hampton Loade, a distance of seven point two kilometres, and we arrived at 9.20. A useful way to represent a journey is to draw a position–time graph, marking positions on the vertical axis, with BN for Bridgnorth and HL for Hampton Loade. The horizontal time axis is divided into equal time intervals, starting at nine o’clock, when the journey began, up to 9.20, when we reached Hampton Loade. The graph of the trip from Bridgnorth to Hampton Loade looks like this. Up until nine o’clock, the train was stationary at Bridgnorth, and so the graph is horizontal. As it speeded up, the graph gets steeper. At about six minutes past nine, the train had settled to a constant speed, for a period of five to six minutes. But then it slowed down to enter the cutting at Sterns, where there is a speed limit. Let’s mark the position of Sterns on the vertical axis. At 9.16, once the train had passed through Sterns, the graph gets steeper again, as the train speeded up, until it finally slowed down as it approached Hampton Loade station, where it stopped at 9.20. The next part of the journey took us beyond Hampton Loade to Highley Station, along another three point three kilometres of single-track railway. There were a number of factors which affected our speed. So, where are we now, Keith?
KEITH SHAW
Well, we’re, we’ve just past Alveley sidings, we’re approaching Highley; we’ve come over Highley Bank, which is quite a steep gradient, and it’s reverse curves, that’s curves in one direction and then the other, which makes for increased friction, which makes it more difficult for the driver, and so we should be approaching Highley soon. Of course, in the opposite direction, the gradient works in the driver’s favour, so he doesn’t – it’s important that he keeps to the speed restriction in that direction as well. Slowing now for Highley, I’ll check the time and – 9.27.
JUDY EKINS
After Highley, there’s Arley Station, a further three point five kilometres up the line – which we reached at 9.36. One kilometre beyond Arley, the line crosses over to the other bank of the river.
KEITH SHAW
Well, that’s 9.40. We’re just crossing Victoria Bridge, which when it was built in 1861 was the largest single-span iron structure ever built at that time, by a firm called John Fowler, and it’s 200 feet, single span. Now on the faster section of the line – it doesn’t have so many slacks on as the earlier part, and also of course by now the engine’s warmed up and things are running much more as they should be.
JUDY EKINS
There’s one more stop before Kidderminster, and that’s in the town of Bewdley. To get to Bewdley station we’ve travelled a total of 20 kilometres from Bridgnorth and arrived at 9.50. The next landmark was the tunnel.
KEITH SHAW
We’re just approaching Bewdley Tunnel now, this another timing point – 9.56. Engines of course whistle when they go in, to warn anyone who’s inside that there’s a train coming,
End transcript: Single-track minders - Part 1
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Single-track minders - Part 1
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Activity 23: Graphing the journey

On a sheet of graph paper, plot a distance-time graph for the journey from Bridgnorth to Kidderminster. (Note that the graph shown on the video is actually a position-time graph, because actual places rather than distances from a single place are shown on the vertical axis.) The map shown on the video is repeated in Figure 54. Note down briefly what information you will need to plot the graph.

Work out the highest and lowest speeds reached on the journey. In what sections of the line do they occur?

What is the average speed for the complete journey?

Make some notes to explain how you made these calculations.

Discussion

Figure 54
Figure 54 Distance-time graph of the train journey

The highest speed is reached on the section between Hampton Loade and Highley. This is where the slope of the distance-time graph is steepest. The 3.3 km section was covered in 7 minutes. So the average speed is 3.3/7=0.47 km per minute, or about 28 km per hour.

The graph indicates that the lowest speed occurs on the section between Bewdley and Kidderminster. Here, the slope of the graph is least. The train covers the 4.5 km in 20 minutes, an average speed of 4.5/20=0.225 km per minute, or 13.5 km per hour.

The complete journey of 24.5 km is covered in 70 minutes. So the average speed is 24.5/70=0.35 km per minute, or 21 km per hour.

Figure 55
Figure 55 A map of the journey from Bridgnorth to Kidderminster from the video
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