Category Archives: Flying

Velocity: Orbital vs. Escape

While thinking about escape velocity recently, I wondered why orbital velocity wasn’t the same as escape velocity. The intuition was: consider an object in a circular orbit around the Earth at speed v. If the object speeds up just a smidge, then its centrifugal force increases, which pulls it slightly further away from Earth, where gravity is weaker, so it goes even further away, etc. It seems like a positive feedback chain reaction, the object getting progressively further away from E. That would imply that orbital velocity equals escape velocity, because if you go even a smidge faster, you’ll eventually escape orbit.

However, I worked out the equations and escape velocity is not equal to orbital velocity, but it’s about 41% faster (actually, square root of 2 faster). Upon further thought, I realized my earlier intuition missed a key point: as the object moving slightly faster goes further from Earth, its trajectory flattens out. When its trajectory is a circle, the force of Earth’s gravity is perpendicular to its motion, so it does not affect the object’s speed. But when the object’s trajectory flattens out, it’s no longer a circle, so Earth’s gravitational pull is no longer perpendicular to its motion. Some small portion of Earth’s gravitational pull is slowing it down! Then, of course, pulls it forward speeding it up as it comes around the other side of the ellipse.

So when the object speeds up a smidge, its orbit becomes elliptical. It has to go significantly faster than that to escape from Earth. In fact, about 41% faster since the difference is the square root of 2.

This also means orbits are stable: if the velocity changes a bit the shape of the orbit changes, but it stays in orbit. If escape velocity equaled orbital velocity, orbits would be unstable: the slightest bump would send it out into space or spiraling inward.

When the math contradicts intuition, it leads to re-thinking which can deepen one’s intuitive understanding.

Escape Velocity

Escape Velocity is commonly described as the minimum speed an object must reach to escape the Earth (or other celestial body) into space. But this definition is ambiguous and can be misleading.

You can escape the Earth at walking speed, if you could walk straight up; you don’t need anywhere near escape velocity. Imagine a rocket launch; in the first few seconds just as it starts to move, it’s going up at walking speed. Theoretically, it could throttle back the engines to maintain that slight upward speed all the way into space, so long as it didn’t run out of fuel or become unstable. A space elevator could also leave Earth at mundane speeds.

The key to this ambiguity is escape velocity applies to a free body, an object that is passively moving according to the laws of physics, having no thrust of its own. In other words, if a rocket achieves escape velocity, it could at that point turn off its engines and it would still escape the Earth. Intuitively it seems the higher the altitude, the slower the escape velocity. This turns out to be correct.

Escape velocity is easy to understand and derive mathematically with some creative thinking. Imagine 2 objects in space (a big one and a much smaller one, like the Earth and a stone) surrounded by vacuum, no other objects. So there is no friction and no other bodies exerting gravitational pull. Suppose the stone is at rest relative to the Earth and almost infinitely far away. The gravitational pull is effectively zero. Imagine the stone precariously balanced just on the outer rim of Earth’s gravity well. Then you nudge the stone just a smidge toward the Earth, so it crosses that rim and the Earth starts pulling on it (and vice versa). It starts out slow, but accelerates toward the Earth incrementally faster and faster.

Eventually, when the stone reaches the Earth it will be moving very fast. Escape velocity is the speed it is going just before it smashes into the Earth. Or if it misses the Earth, it’s the speed at its point of closest approach. More correctly and completely, the stone is always traveling at escape velocity at every moment along its path. The escape velocity for that distance from the Earth, is the speed at which the stone is moving when it’s that far away.

Note: the bold face statement above is the nut of this explanation. When you grok its fullness, you grok the fullness of escape velocity.

That’s because of conservation of energy. When the stone was at the rim of Earth’s gravity well, it had a lot of potential energy. At the point of closest approach, all that potential energy has been converted into kinetic energy. Assuming no atmosphere, no losses, the two energies are equal. So as the stone speeds past the Earth, slowing down due to the same gravitational pull that sucked it in, that kinetic energy is converted back into potential energy. So it must reach the exact same distance away when it peters out and eventually stops.

The direction of motion is irrelevant to escape velocity. Normally this seems counterintuitive, but understanding escape velocity with our theoretical example, you can easily see why direction doesn’t matter. At that point of closest approach, it doesn’t matter what direction the stone is moving relative to the Earth. It could be nearly straight up (can’t be exactly straight up, or it wouldn’t have missed), or nearly horizontal. If it’s going horizontal, it has to travel further to escape, but being horizontal, gravity isn’t pulling it as hard. These conflicting factors are equal and cancel each other. All that matters is the altitude (distance of closest approach), because the speed depends only how much energy it’s gained from Earth’s gravity field.

If, at that point of closest approach, the stone were moving any slower, then it would have less kinetic energy, and it will not go as far away. That means it won’t make it to the rim of Earth’s gravity well, so it will still be inside the well, reverse direction and eventually come back to Earth. So escape velocity is the minimum speed a free body can have, and escape the Earth.

Of course, in the real world direction does matter. The Earth has an atmosphere that creates a lot of friction and energy loss at high speeds. If you go straight up, you’re in the atmosphere for a shorter time, less energy loss. If you go horizontal, you’re in the atmosphere longer and will lose more energy.

Here is the mathematical derivation:

escapeVelocity

Flying over Mountains

Mountain flying in small single engine airplanes adds potential risks. But there are things pilots can do to minimize these risks and do it safely.

Night: don’t do it. Night adds risk, mountains add risk. Either alone can be safe if you take precautions. But don’t combine them.

Fuel: use a bigger fuel reserve because there is greater chance to be delayed – either having to take an alternate route, or due to strong wind. But don’t simply top off the tanks if you don’t need to because that can add unnecessary weight, impairing performance. Flying in mountains at high DA, you need all the performance you can get. Normally, I plan to land with at least 1 hour of fuel in the tanks. When flying in mountains I increase this to 90 minutes.

Water: you’ll be at high altitude which is dehydrating. Bring plenty of water for everyone on board.

Wind: understand how wind flows around mountains and plan accordingly. Here’s a great picture I got by Googling:

wind-mountain

  • I’ve circled in red the areas you should never fly. Here be dragons: down-drafts and turbulence.
  • Nasty wind effects begin at around 20 kts. If the UA forecast shows winds this strong over the mountains you’re crossing, consider an alternate route or be at least 50% higher than the highest ridge (see below).
  • Crossing mountains against the wind is super nasty:
    • First, you have the obvious: a headwind which slows down your ground speed and kills efficiency – and you’re at high altitude where winds are generally stronger.
    • Second, the downwind side of the mountain has strong down-drafts, so you need to climb just to maintain level altitude, which saps speed and efficiency even further.
    • Third, the downwind side also has a lot of turbulence. The last time you need turbulence is when your ground speed is slow and you are fighting down-drafts.
    • Fourth, you are at high altitude, where your engine has less power and your prop & wing are less efficient. Just when you need power and efficiency the most, you don’t have them.
  • Nasty wind effects exist from the surface to about 50% higher than the height of the ridge. For example if you’re crossing an 8,000′ mountain on a windy day, you need to be at least 12,000′ high to avoid the worst down-drafts and turbulence, though you will still feel some mountain wind wave effects.
  • Climb early. Know how high you need to be and get there well before you reach the mountains. If you get too close before climbing, you may get stuck in down-drafts making it impossible to climb to the altitude you need.

To summarize, if you’re flying over mountains here are ways to minimize the risk:

  • Do it during the day.
  • Have more fuel than you need, but not so much you’re unnecessarily heavy.
  • Climb to 50% higher than the highest ridge at least 50 miles before you reach it.
    • This is not necessary with calm or very light winds.
  • If you’re going against the wind, use full power properly leaned. At 10,000′ your engine only makes about 70% of its rated power. You need it all, and in the high altitude cool thin air you can’t hurt the engine.
  • If you get into mountain waves, ride them instead of fight them.
    • If they’re smooth and gentle, let them push you up or down at least 500′ before counteracting them.
    • If they’re pushing you too far up, maintain power, nose down & gain some airspeed.
    • If they’re pushing you too far down, maintain power, nose up to Vy if necessary to minimize the altitude loss.
    • Remember at sea level Vx is always slower than Vy. As you climb Vx gets faster, Vy gets slower, until they meet, which is your airplane’s absolute ceiling.
  • If you’re going against the wind, after you reach the top of the highest ridge you’ll be in an up-draft with a lot of altitude you no longer need. Nose down and convert all this energy into airspeed. You’ll regain some of the lost time and efficiency.

P-51 Flight Experience

On July 4, 2016, I flew in the P-51C. Got a little instruction and stick time, and a 0.7 hour log book entry I will treasure for the rest of my life. It was every bit as amazing and cool as I expected.

After a few mins getting the feel of the P-51, this conundrum hit me. Sure it has enough fire-breathing power and performance to kill you in a blink, and there’s this foreboding sense of responsibility not wanting to screw up and destroy a magnificent 70-year-old piece of history despite having the CFI on board doing the real flying. Yet at the same time I expected the P-51 to be this beasty monster and instead found it so smooth and responsive and natural feeling, not just fast but also quick despite its weight, sensitive but not twitchy, just a really sweet flying airplane. I don’t mean to imply it’s easy to fly, at 0.7 hours I’m no judge of that, I don’t even have the skill to fly it on my own – the CFI up front did all the hard work. Even so, it seemed much more well behaved than I expected which left me in even greater awe of the men who designed it, and admiration for the men who mastered this airplane and flew it into battle.

From what I’ve read, more than half the pilots and planes lost during WW-II were in training or other non-combat activities. That goes to show that flying warbirds like the P-51 and AT-6 is like holding a tiger by the tail. I expected the tiger to be evident in every aspect of the plane: noise, control, handling, vibration, twitchiness, etc. What I didn’t expect was such a smooth responsive sweet flying airplane. Yet this actually makes it even more scary and dangerous because the tiger hides and whispers “you can do it”, until it suddenly strikes.

Photos & videos here.

How Strong are Small Airplanes?

The FAA defines 3 categories for small airplanes:

Normal: all standard private and commercial flight maneuvers up to 60* bank angle. Must withstand 3.8 G or more.

Utility: additional flight maneuvers like spins and > 60* bank angles. Must withstand 4.4 G or more.

Acrobatic: any maneuver or bank angle not prohibited by the POH. Must withstand 6.0 G or more.

All certified GA airplanes meet the Normal category, many (like the Cessna 172) meet the Utility category, and some meet Acrobatic. With 3.8 G as the minimum, this means airplanes are built very strong.

You don’t really know how strong the airframe is because the G rating is a minimum. It can handle that G load under normal operation. Certainly it can handle more, but how much more is unknown. If you exceed it, you’re the test pilot – that’s bad, don’t do that.

Being certified Utility doesn’t necessarily mean the airplane can perform any maneuver exceeding 60* of bank. For example many aircraft certified Utility are not approved for spins. Prohibitions like this are listed in the POH.

Airplanes certified for multiple categories may not always satisfy all categories. For example the Cessna 172 is certified Utility only when gross weight is under 2,100 lbs. and CG is forward of a certain point. Otherwise, it’s certified Normal.

Most Common GA Accident Types

In the past 100 years nobody has created a new way to destroy an airplane. Here are the most common ways, roughly in order of most common first:

  1. Weather: pilot didn’t respect Mother Nature (she doesn’t have to respect you – she was here first).
  2. Fuel: pilot ran out of fuel (airplane engines run better with fuel).
  3. Planning: or lack thereof – over gross weight, out of CG limits, density altitude, VFR into IMC, etc.
  4. Maintenance: pilot departed with known aircraft deficiency (airplanes work best when properly maintained).
  5. Pilot was cognitively impaired (fatigue, drugs, etc.).
  6. Stupidity: pilot intentionally did something stupid (buzzing, “watch this”, etc.).

Every aviation accident I know of falls into at least one of these categories – sometimes more than one. The good news is, improving safety is simple common sense. Don’t do these things! Safety improves one pilot at a time. If you don’t do these things, you’ve improved your safety roughly 10-fold and you’re making GA safer than driving or bicycling.

Power Moves the Airplane

Actually it’s thrust that moves the airplane, but power determines how much thrust you can get any any given airspeed.

Thrust is what pushes (or pulls) an airplane. Thrust is related to power by a simple equation. Removing constant factors, Power = Thrust * Speed. An engine with a constant power output – like any piston engine – gives max thrust (thus max acceleration) at low speeds, with less and less thrust (thus diminishing acceleration) as speed increases, until you reach top speed, where thrust = drag and acceleration is zero.

Put differently: if someone says, “My engine produces 250 lbs. of thrust” you really know nothing. Why? No engine produces the same thrust at all speeds, and he hasn’t told you at what speed it produces that thrust. But if he says, “My engine produces 250 horsepower” you know everything you need to know. From this you can compute how much thrust it produces at any speed, so given the airplane’s mass and drag coefficient, you can compute its acceleration, climb rate and top speed. In this sense, power is what moves the airplane – it determines how much thrust you can get at any speed.

Another way to think about this: forces move objects. But force alone doesn’t tell us much. Any object that actually moves in the real world, moves a certain distance in a given amount of time. Force * distance is work, and work over time is power.

Here’s another example why Power is what moves things – not force or work: Suppose someone tells you, “My truck has 500 ft. lbs. of torque”. From this it’s impossible to know how fast it can tow a load up a hill. Towing the load up the hill takes a certain amount of work. You can do that work fast or slow. Power is how quickly you do the work. The little electric motor that rolls the truck’s windows up and down can also tow any load up a hill – given enough time, it can do any amount of work. As Archimedes said, “Give me a lever long enough and I’ll move the world”. If you want to tow a real load up a real hill in the real world, you care how long it takes. Power tells you that. If that truck makes that 500 ft.lbs. at 3,000 RPM, it has 3 times as much power as it does if it makes the same torque at 1,000 RPM. And it can tow the load up the hill 3 times faster. If he told you, “My truck has 500 horsepower”, you compute how fast it can tow the load up the hill. Once you know that, it doesn’t matter how much torque it has or at what RPM it makes the torque. Torque * RPM = Power, and any combination of Torque and RPM that makes that power will do the job.

In short: because the airplane moves an actual distance over an actual time in the real world, power is what moves the airplane.

More precisely, Power = Thrust * Speed * Efficiency. The reason we must include efficiency will become clear later.

First, consider an airplane with a fixed pitch prop. At full throttle standing still it pulls around 2300 RPM – well shy of redline. It must be designed this way because otherwise, it wouldn’t be able to fly fast enough. As the airplane starts moving, the prop blades see reduced angle of attack to the oncoming air, which reduces resistance to motion, it would want to spin faster but couldn’t because it would already be at redline. The pilot would have to gradually pull back the throttle during the takeoff roll and climb to avoid over-revving.

Now 2300 RPM is about 85% of redline, and since Torque * RPM = Power, the engine is making 85% of its rated power. If you have a 160 HP engine then you have about 135 HP during the takeoff roll, with the engine at full throttle pulling 2300 RPM.

Propeller efficiency is a key factor – it determines how much of the power the engine is making, is converted into thrust. The rest of the power is converted into noise and turbulence. A propeller achieves its ideal efficiency only at a certain angle of attack. This amounts to a medium-ish airspeed. At slow speeds, and at fast speeds, the propeller is less efficient. So our airplane in the above example effectively has less than 135 HP because it’s moving slowly and the prop is gaining efficiency as it speeds up. To be clear, the engine is making 135 HP but some of that power is being converted into turbulence and noise instead of thrust, so there is less than 135 HP making thrust.

So theoretically, the speed of maximum thrust is zero. That comes directly from Power = Thrust * Speed. But in reality, the speed of maximum thrust is higher. In reality, as you go down the takeoff roll, both power and thrust are increasing because the prop is gaining efficiency as you gain speed.

Now let’s consider 2 key airspeeds: Vx and Vy. Every pilot knows Vx is the airspeed that gives greatest angle of climb. If you need to clear trees on the takeoff roll, fly at Vx. Vy is the airspeed that gives the greatest rate of climb. If you want to climb to 10,000 feet as quickly as possible, fly at Vy. The difference between Vx and Vy is thrust vs. power. That is, Vx is the speed of maximum excess thrust, and Vy is the speed of maximum excess power. Here, excess means, above the amount needed to sustain level flight. Vx is always slower than Vy.

One way to think about this, is every climb has an angle and a rate of altitude gain. The angle is determined by excess thrust beyond what is needed for level flight. The rate of altitude gain is determined by excess power beyond what is needed for level flight. So, max excess thrust gives the biggest angle and max excess power gives the highest rate.

Recall the drag vs speed curve of an airplane in flight. Induced drag is high at low speeds, low at high speeds. Parasitic drag is the opposite. The speed having least total drag is the point where they are equal. This is usually much slower than you would normally fly. In my 172, it’s about 60 knots. This is also the speed at which you can glide the longest distance: called Vldmax. Now, knowing that Vx is the speed at which you have the most excess thrust, and drag is what saps your thrust, you might expect Vx to equal the speed of minimum drag. It’s close to that, but always slower. Why? Because your engine makes more thrust at lower speeds, and the relationship is linear. As you slow down just a bit from Vldmax, total drag increases less than linearly, while thrust increases linearly. This means thrust increases more than drag, giving you more excess thrust. Drag is increasing exponentially as you slow down (or speed up) from Vldmax, so if you slow down even more, drag will increase more than thrust.

Now consider Vy, the speed where you have the most excess power. This is a little more complex than Vx because it depends on efficiency. First let’s derive the airspeed of minimum power. It is always slower than the airspeed of minimum drag. Let’s start from that speed – Vldmax – and find out why. Suppose it takes power P to fly at Vldmax. How much power would it take to fly just a bit slower? Flying slower, the drag increases a little – but less than linear – so our speed dropped more than our drag increased. Drag = Thrust and Power is Thrust * Speed, and we just saw that speed decreased more than thrust increased, which means their product, power, is smaller. Thus the airspeed requiring minimum power is slower than the airspeed of minimum drag. However, if we keep slowing down, drag will increase rapidly and it will require more power – not less!

Here is where propeller efficiency enters the picture. It’s simple and obvious that if power output is constant, then the speed of maximum excess power is equal to the speed of minimum power required for flight. That would be slower than Vldmax, yet we know Vy is higher than Vldmax! But in reality, power output is not constant because propeller efficiency is not constant. Its efficiency peaks at a higher airspeed than Vldmax. Thus when you’re at Vldmax, you have poor propeller efficiency. If you speed up, your propeller becomes more efficient, and the gain in efficiency is greater than the increased power required to fly at the higher airspeed. The speed of maximum excess power is always somewhat higher than Vldmax.