Angle of climb/rate of climb
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To further clarify this, an excellent depiction of these relationships is presented on pages 131 & 132 of this book. Note the Vx and Vy lines on the graph.
Flight testing of fixed-wing aircraft - Ralph D. Kimberlin - Google Books
Flight testing of fixed-wing aircraft - Ralph D. Kimberlin - Google Books
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BEST ANGLE VS BEST RATE OF CLIMB.
BEST ANGLE OF CLIMB ALSO KNOWN BY THE HEAVY METAL AS BEST GRADIENT
SPEED THAT;
GIVE THE AIRFRAME MOST POWER AVAILABLE OVER POWER REQUIRED.
SAY THE SPEED IS 150 KTS FOR BEST GRADIENT.
THIS EQUATES TO A SPEED WHERE THE LEAST AMOUNT OF POWER IS REQUIRED TO FLY AT 150 KTS, AND ANY POWER AVAILABLE OVER THIS MIN POWER REQUIRED CAN BE USED TO GAIN HEIGHT.(increase ange of climb)
SAY 150 KTS NEED 40% OF THE POWER AVAILABLE, THE OTHER 60% IS WHAT IS AVAILABLE TO CLIMB.
THE OTHER SIDE OF THE ARGUMENT SAY 180 KTS GIVES THE BEST RATE OF CLIMB IN UNITS PER MINUTE BUT IT WILL NEED MORE POWER TO MAINTAIN 180 KTS BUT THE LD RATIO IS BETTER SO THE AIIRCRAFT WILL CLIMB AT A GREATER NUMBER OF UNITS PER MINUTE SO BE HIGHER THAN CASE A AFTER A FIXED TIME; BUT FURTHER DOWN TRACK., WHICH WOULD NOT AVOID A NASTY NOISE IF THERE WAS A NEAR OBSTACLE.
SPEED THAT;
GIVE THE AIRFRAME MOST POWER AVAILABLE OVER POWER REQUIRED.
SAY THE SPEED IS 150 KTS FOR BEST GRADIENT.
THIS EQUATES TO A SPEED WHERE THE LEAST AMOUNT OF POWER IS REQUIRED TO FLY AT 150 KTS, AND ANY POWER AVAILABLE OVER THIS MIN POWER REQUIRED CAN BE USED TO GAIN HEIGHT.(increase ange of climb)
SAY 150 KTS NEED 40% OF THE POWER AVAILABLE, THE OTHER 60% IS WHAT IS AVAILABLE TO CLIMB.
THE OTHER SIDE OF THE ARGUMENT SAY 180 KTS GIVES THE BEST RATE OF CLIMB IN UNITS PER MINUTE BUT IT WILL NEED MORE POWER TO MAINTAIN 180 KTS BUT THE LD RATIO IS BETTER SO THE AIIRCRAFT WILL CLIMB AT A GREATER NUMBER OF UNITS PER MINUTE SO BE HIGHER THAN CASE A AFTER A FIXED TIME; BUT FURTHER DOWN TRACK., WHICH WOULD NOT AVOID A NASTY NOISE IF THERE WAS A NEAR OBSTACLE.
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The best angle plane will be moving more slowly across the ground, so combined with the steeper angle, will clear an obstacle that the best rate plane might not.
I seem to have got them back to front. Best rate is shallower/flatter, best angle is steeper.
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Its probably worth remebering to emphasise to students (at PPL stage anyway) that if they are in a situation where the difference between best gradient climb and best rate of climb matters for terrain clearance... ...well they really shouldn't be there 
H
H
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...well they really shouldn't be there
If that pilot has recognized a need to outclimb rising terrain, best angle is likely most appropriate. If outclimbing changing weather, or traffic is necessary, and terrain is not a factor, best rate is probably more appropriate.
If you expect to climb for a long period, best rate will be much better for cooling.
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I seem to have got them back to front. Best rate is shallower/flatter, best angle is steeper.
If that pilot has recognized a need to outclimb rising terrain, best angle is likely most appropriate.
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If you're looking for an explanation as to why max excess thrust will give max angle of climb and max excess power will give max rate of climb, you need to go over a TON of basics. Thrust and power are connected somewhat like torque and power. I wrote an article on torque and power that you can check out here: http://www.box.net/shared/tpby1m893e6mipiv6sjp
When you talk about climbs you have to discuss power required and power available and thrust required and thrust available. I find it's best to start with power when discussing reciprocating engines and start with thrust when talking about jet engines. It can get incredibly confusing so take it one step at a time. The Advanced Pilot's Flight Manual has some good stuff on climbs that I suggest you read. You've asked such a broad question it's impossible to answer on here. I could write a book on it. Once you've read that and understand it a bit, I can help you with specific questions.
When you talk about climbs you have to discuss power required and power available and thrust required and thrust available. I find it's best to start with power when discussing reciprocating engines and start with thrust when talking about jet engines. It can get incredibly confusing so take it one step at a time. The Advanced Pilot's Flight Manual has some good stuff on climbs that I suggest you read. You've asked such a broad question it's impossible to answer on here. I could write a book on it. Once you've read that and understand it a bit, I can help you with specific questions.
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Dboy: just to confuse you a bit more:
sometimes when flying at max rate speed we choose best angle speed to increase our actual rate of climb over a period of time
Explanation/practical example: You are climbing at mach 0.76 and the atc ask you to expedite. You select max rate which is 266 KTS indicated speed today.
You are asked again to reach your level as soon as possible, you then select max angle (255 today?) which increases your actual angle AND your actual rate at the same time during the speed loss!
Let's go even further: you've just reached you best angle speed at 1200 feet below your level and the ATC is wishing you make it even faster, due to converging traffic (bad example).
You select then minimum speed for one engine out, maybe 242 today, (not a good procedure, but the result remains true for the subject of this thread): you find yourself increasing your actual angle AND your actual rate while the speed is decreasing.
Now you are reaching your level meanwhile you've just reached your minimum one engine speed, which is below both your max rate speed and best angle speed but have increased your actual max rate and actual best climb by doing so...
Hope I have confused you enough
sometimes when flying at max rate speed we choose best angle speed to increase our actual rate of climb over a period of time
Explanation/practical example: You are climbing at mach 0.76 and the atc ask you to expedite. You select max rate which is 266 KTS indicated speed today.
You are asked again to reach your level as soon as possible, you then select max angle (255 today?) which increases your actual angle AND your actual rate at the same time during the speed loss!
Let's go even further: you've just reached you best angle speed at 1200 feet below your level and the ATC is wishing you make it even faster, due to converging traffic (bad example).
You select then minimum speed for one engine out, maybe 242 today, (not a good procedure, but the result remains true for the subject of this thread): you find yourself increasing your actual angle AND your actual rate while the speed is decreasing.
Now you are reaching your level meanwhile you've just reached your minimum one engine speed, which is below both your max rate speed and best angle speed but have increased your actual max rate and actual best climb by doing so...
Hope I have confused you enough
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KAG, what you're describing is called a zoom climb. It shouldn't be related to holding a Vx speed for a limited time. A zoom climb is not sustained like Vx and Vy are. During sustained climbs, mechanical work is being transferred into potential energy (altitude). In a zoom climb, it's typically done at full thrust/power and a majority of the aircraft's climb performance is from it's kinetic energy being converted into potential energy (altitude).
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You got it right.
However if we want to be accurate, even if I agree with what you mean, NO climb speed, including best angle and max rate are sustainable on our aircrafts...
A "zoom climb" as you call it is sometimes the real answer to an emergency (wind shear at very low altitude, stall at very low altitude, imminent crash...) to avoid terrain, best angle speed won't be the one we will be looking for in those cases, the procedure will be to pull up to stick shaker (close to stall speed) to get the actual "best angle" and save your life, because the best angle speed remains in the real life a theorical speed that doesn't fit all real situations requesting the actual best angle performances from your airplane.
A zoom climb is not sustained like Vx and Vy are.
A "zoom climb" as you call it is sometimes the real answer to an emergency (wind shear at very low altitude, stall at very low altitude, imminent crash...) to avoid terrain, best angle speed won't be the one we will be looking for in those cases, the procedure will be to pull up to stick shaker (close to stall speed) to get the actual "best angle" and save your life, because the best angle speed remains in the real life a theorical speed that doesn't fit all real situations requesting the actual best angle performances from your airplane.
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However if we want to be accurate, even if I agree with what you mean, NO climb speed, including best angle and max rate are sustainable on our aircrafts...
Now you're using a different meaning of "sustainable". I agree with your statement about zoom climbs but they shouldn't be referred to as "best angle" climbs. That's incredibly confusing especially when trying to teach a student what best angle (Vx) climbs are. Discussing the benefits of the zoom climb, like you did, is good but it shouldn't be called Vx (best angle) because that is something different.
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Why complicate everything?
Best angle is thurst minus drag over weight and best rate is proportional to the amount of excess HP available over that required to fly the aircraft straight and level.
Instructors should teach people to fly not bore them to death trying to be clever.
Best angle is thurst minus drag over weight and best rate is proportional to the amount of excess HP available over that required to fly the aircraft straight and level.
Instructors should teach people to fly not bore them to death trying to be clever.
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Why complicate everything?
Can you answer this question?... If a piston airplane's only way of moving is controlled by how much thrust is being created, then what does power have to do with it? Consider a fixed pitch, single engine airplane. Oh, and don't complicate it!
EDIT: Complicate is a very ambiguous word. To someone who just passed grade 5 math it would probably be considered complicated but to someone who has a PhD in aeronautical engineering it could be considered overly simple. What level of education should pilots be expected to have?
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I agree with your statement about zoom climbs but they shouldn't be referred to as "best angle" climbs. That's incredibly confusing especially when trying to teach a student what best angle (Vx) climbs are. Discussing the benefits of the zoom climb, like you did, is good but it shouldn't be called Vx (best angle) because that is something different.
A bit less focus on the physics and more concern about the reality and actual use of best angle concerning the actual operation of an airplane might be necessary to avoid speaking about matter of very little use when facing an emergency or having to deal with acceleration/deceleration change of flight level in a busy airspace.
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KAG... This is really simple but obviously not obvious. If two words or sentences or phrases are spelt differently, they're NOT the same! A zoom climb is a zoom climb. Best angle of climb is best angle of climb. They should be discussed separately. Just like Vy and Vx both have a rate of climb and an angle of climb, so too does a zoom climb. And there are certain circumstances where one climb offers more performance than the others. That should be discussed, but it doesn't change the fact that they're different.
That statement is entirely irrelevant to what I've said. I never said that Vx should be used in all cases where you need to avoid an obstacle. I will say that if you're required to use a zoom climb to clear an obstacle without crashing then you're in a place you never should have been in. You should get a slap on the wrist for that.
A bit less focus on the physics and more concern about the reality and actual use of best angle concerning the actual operation of an airplane might be necessary to avoid speaking about matter of very little use when facing an emergency or having to deal with acceleration/deceleration change of flight level in a busy airspace.
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I will say that if you're required to use a zoom climb to clear an obstacle without crashing then you're in a place you never should have been in.
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I quite agree, over the last 28 years I have produced some crappy pilots but all excellent commanders.