Q17. Bit of a surprise this one!
It may just be that you used a wrong word carelessly, but I am going to assume that you wrote what you meant- in which case it reveals a misunderstanding which I'll now attempt to correct. However I'd be interested to know if there is anything in the Study Pack etc that gave you the wrong impression, so that I can attempt to cure the problem at source for others.
Headway is not about safe separation of trains. Think about it; I am sure that you know that in 3 aspect signalling a train can be just clear of the overlap joint of one signal (at red) and a train can be safely approaching the signal in rear (at yellow). There is at least braking distance between the signals so the second train ought to stop at the red signal and there would still be an additional safety distance- the overlap- between the trains.
Your answer to Q18 states that for trains to be at minimum non-stop headway there must be more than two signal sections between them. This is indeed correct; the driver of the second of the two trains must see a Green- i.e. they can drive their train in a manner unaffected by the presence of the first train. Hence headway is all about capacity ; if the trains become closer that the minimum headway separation, then the second of them will encounter cautionary aspects and therefore it must brake, therefore lose time compared to the timetable and indeed waste line capacity by not using to the optimum for the speed for which it is signalled.
I think it is best to think of headway as a TIME; it only equates to a DISTANCE on the assumption of constant speed. Where the speed of the line fluctuates (as all really do), or where trains need to slow to pass over pointwork, or indeed slow and stop at a station, a pair of trains doing the same thing will remain a constant headway time apart ( the time interval between them measured at any site along their route is the same), but NOT a constant distance apart. Imagine train 1 slowing for a station whilst train 2 behind it will be travelling at full speed- their distance separation will be reducing. When train 1 is stopped at the station, train 2 will be catching it up fast. However train 1 will eventually accelerate to maximum speed again whilst train 2 will itself need to slow down for the station; hence their separation will then be increasing again. Since the trains do identical things, then once both are travelling at full speed again, their distance separation will return to its original value. This is obviously a key difference; "headway distance" can be a useful concept for non-stop, but is meaningless otherwise.
Your last para is tolerably right but, as above, measurement in metres is only sensible when placing signals on constant speed running line. Measurement in seconds is far more useful and is what mainline railways tend to use. Measurement in trains per hour is generally used for metros. The rationale is when a line is used at maximum capacity then it is the total number of trains that the "pipe" can accommodate which is important (as you say for the TOC). On those lines that are not so heavily used then this is not a useful measure, but what is far more significant is the minimum time interval between successive trains that is important.
Take Paddington; a TOC wants to run an HST to Bristol, one to South Wales and then one the West Country leaving at 5 minute intervals in a "flight" and then no more until the next half hour. The remainder of the time within the hour is used for slower trains- this maximises the use of the line. There is a mix of traffic and capacity is lost when two trains of different speeds are timetabled adjacent to each other, which is why the "flighting" of similar trains is economical of capacity. "Trains per hour" is only a useful measure when all trains are "the same".
Another rather different example; Machynlleth. Generally a 4 car train arrives from Birmingham with one driver (for economy) and it then splits into a 2 car set going north to Pwllheli and one going the other way along the coast to Aberystwth with a separate driver. The few miles between the station and the subsequent junction needs to be signalled to permit two relatively closely following trains- or else the passengers in the second train would have a long delay on their through journey. However there may well not be another train along the line in that direction for another 2 hours; you can see that "trains per hour" is a meaningless concept here, but that there i a need to signal that line with a headway of only a few minutes.
Q18/19.
Look at what you wrote again; can you see your mistake.
Hint: I don't think you a Physicist- your dimensions are wrong!
It may just be that you used a wrong word carelessly, but I am going to assume that you wrote what you meant- in which case it reveals a misunderstanding which I'll now attempt to correct. However I'd be interested to know if there is anything in the Study Pack etc that gave you the wrong impression, so that I can attempt to cure the problem at source for others.
Headway is not about safe separation of trains. Think about it; I am sure that you know that in 3 aspect signalling a train can be just clear of the overlap joint of one signal (at red) and a train can be safely approaching the signal in rear (at yellow). There is at least braking distance between the signals so the second train ought to stop at the red signal and there would still be an additional safety distance- the overlap- between the trains.
Your answer to Q18 states that for trains to be at minimum non-stop headway there must be more than two signal sections between them. This is indeed correct; the driver of the second of the two trains must see a Green- i.e. they can drive their train in a manner unaffected by the presence of the first train. Hence headway is all about capacity ; if the trains become closer that the minimum headway separation, then the second of them will encounter cautionary aspects and therefore it must brake, therefore lose time compared to the timetable and indeed waste line capacity by not using to the optimum for the speed for which it is signalled.
I think it is best to think of headway as a TIME; it only equates to a DISTANCE on the assumption of constant speed. Where the speed of the line fluctuates (as all really do), or where trains need to slow to pass over pointwork, or indeed slow and stop at a station, a pair of trains doing the same thing will remain a constant headway time apart ( the time interval between them measured at any site along their route is the same), but NOT a constant distance apart. Imagine train 1 slowing for a station whilst train 2 behind it will be travelling at full speed- their distance separation will be reducing. When train 1 is stopped at the station, train 2 will be catching it up fast. However train 1 will eventually accelerate to maximum speed again whilst train 2 will itself need to slow down for the station; hence their separation will then be increasing again. Since the trains do identical things, then once both are travelling at full speed again, their distance separation will return to its original value. This is obviously a key difference; "headway distance" can be a useful concept for non-stop, but is meaningless otherwise.
Your last para is tolerably right but, as above, measurement in metres is only sensible when placing signals on constant speed running line. Measurement in seconds is far more useful and is what mainline railways tend to use. Measurement in trains per hour is generally used for metros. The rationale is when a line is used at maximum capacity then it is the total number of trains that the "pipe" can accommodate which is important (as you say for the TOC). On those lines that are not so heavily used then this is not a useful measure, but what is far more significant is the minimum time interval between successive trains that is important.
Take Paddington; a TOC wants to run an HST to Bristol, one to South Wales and then one the West Country leaving at 5 minute intervals in a "flight" and then no more until the next half hour. The remainder of the time within the hour is used for slower trains- this maximises the use of the line. There is a mix of traffic and capacity is lost when two trains of different speeds are timetabled adjacent to each other, which is why the "flighting" of similar trains is economical of capacity. "Trains per hour" is only a useful measure when all trains are "the same".
Another rather different example; Machynlleth. Generally a 4 car train arrives from Birmingham with one driver (for economy) and it then splits into a 2 car set going north to Pwllheli and one going the other way along the coast to Aberystwth with a separate driver. The few miles between the station and the subsequent junction needs to be signalled to permit two relatively closely following trains- or else the passengers in the second train would have a long delay on their through journey. However there may well not be another train along the line in that direction for another 2 hours; you can see that "trains per hour" is a meaningless concept here, but that there i a need to signal that line with a headway of only a few minutes.
Q18/19.
Look at what you wrote again; can you see your mistake.
Hint: I don't think you a Physicist- your dimensions are wrong!
PJW