# Constant Acceleration Versus Coasting Conundrum

1. Apr 9, 2016

### Dr Wu

A disclaimer: I'm trying to limit myself to asking questions on Physics Forums, partly because I feel I've gone way past my quota. Still, there is one question that simply won't go away, and it concerns the relative efficiencies between two forms of motion. The first involves continuous acceleration/deceleration; the second a far shorter, brisker spell of acceleration/deceleration linked by a period of inertial motion. Three assumptions must hold true: (1) distance (2) flight-time and (3) fuel use remain precisely the same in both modes of travel.

There's quite a lot of info about this subject on the net, but somehow I keep getting left behind. Also, there's a marked tendency to bring in Special Relativity, which adds a layer of complexity which I really can do without right now. Newton's Laws of Motion are plenty enough to be getting on with, at least for me. I've chosen space travel just to keep things as simple as possible. I've also tried working this out on the calculator pad - so much so that I don't understand anything at all now.

(Again) Many thanks.

2. Apr 9, 2016

### Megaquark

Since you're assuming the same distance, time, and fuel usage in both situations, they would both be just as efficient.

3. Apr 9, 2016

### billy_joule

The answer is - 'it depends'.
Consider a petrol car, the efficiency of it's engine varies with engine speed and throttle position, generally efficiency increases with both.
So a petrol engine is more efficient when under heavy acceleration (wide open throttle, high rpm) than light acceleration (partial throttle, low rpm).
So to get better fuel economy the engine must be kept closer to it's peak efficiency point, that is, use heavy acceleration. ( This is the approach hypermilers (extreme fuel economy enthusiasts) use).

A space craft doesn't use a petrol engine but rocket engines also have peak efficiency points, so whichever approach keeps the rocket closer to it's peak efficiency point will get better fuel economy. If you're not using a rocket then it depends on the efficiency characteristics of whatever engine you use.

4. Apr 10, 2016

### SteamKing

Staff Emeritus
There's a quota on how many questions you can ask at PF? Since when?

5. Apr 10, 2016

### OldYat47

Flight time implies aircraft. Let's start with that assumption. We'll ignore relative wind effects because the aircraft will fly through them in either case. Whenever aircraft coast they lose altitude and forward speed. The faster they travel, the quicker they lose speed (increased air resistance). That's a lot of energy that has to be replaced by subsequent acceleration. In practice there is a range (somewhat narrow) of engine RPMs that produce the optimum in maintaining altitude and forward speed. It always works out to some narrow range of constant air speed. Too slow and the plane has to adopt a steeper angle relative to the direction of travel, increasing drag. Too fast and aerodynamic drag increases too much.

In cars, take a look at how modern cars with CVTs are programmed. Rarely will you ever see anything close to WOT conditions from the engine.

And no, hypermilers DO NOT use heavy acceleration unless they want to speed up. Accelerate/coast/accelerate/coast is very costly in fuel efficiency. They drive at as constant a speed as possible, coasting where practical. I'm pretty good at it, getting close to EPA mileage in the summer with the A/C going.

And no, Otto cycle engines are not most efficient at WOT. That's thermodynamic theory. In practice internal friction from a variety of sources quickly overcomes reduced pumping loads with a partially closed throttle.

6. Apr 12, 2016

### Dr Wu

I restricted myself to space flight in order to avoid frictional, gravitational and air resistence issues. It seems evident to me (now at least) that increased acceleration, while shortening a given flight-time, incurs the penalty of greater fuel use - although I do take onboard the proviso that certain engines may perform more efficiently at a higher work rate than a lower one. One unknown, of course, is applying this principle to an unknown quantity like (say) a nuclear fusion rocket. Supposing propellent wasn't an issue, I would imagine that such an engine would be several magnitudes more efficient than our current chemical powered rockets, although this is just a conjecture on my part.

As for the continuous acceleration/deceleration v accel/coast/decel conundrum, here's the results for some recent back-of-the-envelope calculations of mine.

Space Probe A (Mass 5 Tonnes) accelerates at 1.0g from zero to a velocity of about 1,900 km/s. This boost lasts for 54 hours, during which time it has covered a distance of 1.25AU. It then coasts for the next 543 hours, covering a distance of 2.5AU. It then decelerates at the same rate it did while accelerating. The totals are as follows: Distance = 5AU. Time = 650 hours (27 days). Energy requirement = 0.018 EJ.

Space Probe B (Mass 5 Tonnes) accelerates at 0.5g from zero to a velocity of about 1,900 km/s. This burn lasts for 217 hours, during which time the probe has covered a distance of 2.5AU. It then turns round and decelerates at the same 0.5g rate. The totals here are: Distance = 5AU. Time: 434 hours (I8 days). Energy requirement: 0.018 EJ.

Summary: distance and energy remain the same. What is different, however, are the two flight-times. Space Probe B takes 66% more time to cover the 5AU distance than Space Probe A, despite having the same energy needs. Conclusion: Space Probe B is clearer the winner in terms of time saved - i.e. you get more bangs for your bucks with constant acceleration/deceleration.

I've yet to see if it's possible to replicate Space Probe B's stats using the accel/coast/decel flight method. This will be my next project.

Many thanks again :)

7. Apr 12, 2016

### Dr Wu

Correction: Space Probe A takes more time to complete the journey, NOT Space Probe B! My apologies.

8. Apr 12, 2016

### billy_joule

One thing to consider is that the payload of craft B will be bigger as it's engine will have less mass.

9. Apr 13, 2016

### devecseri

I've been wondering about that for so long! I knew intermittent speeding was good somehow, but how does it affect fatigue life?