Top E Planes

[FLS]

As you see in F=ma, gravity isn't part of the equation. F=ma, even in a zero gravity enviornment. The force you speak of being calculated with that formula is inertia. Galileo was right.

The a is acceleration and refers to gravity.

[Badboy]

  Nope, they would hit at different times, but not because of their mass, because of the difference in air resistance.  Acceleration due to gravity is 32ft per second, regardless of mass.  But greater air resistance will cause one to fall slower.

Nope, with the same air resistance the one with the greater mass will fall more quickly. If you take two identical hollow balls and fill one with lead and the other with water so that the drag is the same for both, the ball filled with lead will accelerate more quickly as indicated by the equation derived earlier:

a = g - d/m

What happens is that as the mass gets bigger the -d/m term gets smaller thus allowing the ball to fall closer to the acceleration due to gravity and thus faster than a lighter object. As the mass gets smaller the -d/m term gets bigger reducing the acceleration, until eventually if the mass gets low enough the object would float down very slowly.

For objects with the same air resistance the accelerations will be different and the reason is due to their different mass, not different air resistance.

If you take to objects, with differing mass, but equal air resistance, they will both fall at the same speed.

Nope, the heavier object will fall more quickly as indicated by the equation of motion derived earlier.

Regards

Badboy

[Badboy]

Badboy if we consider air resistance don't we have to consider the different coefficients of drag in different aircraft too?

That depends what you are trying to do. If you want to compare how two different aircraft will behave in a dive or a zoom then you need to consider everything.

However, if you are only trying to see how one particular thing will influence the performance, it helps to hold everyhing else constant if possible to get a true comparison. So to see how weight influences a zoom climb it might be helpful to consider two identical aircraft that differ only in weight.

Regards

Badboy   

[Mister Fork]

Top E planes are in two categories. You have aircraft that accelerate the best have either very big engines (like a LA-7) and those with the low drag coefficient ratios and a mid-power engine (like a P-51 Mustang).  We have two completely different things happeneing in both aircraft categories.

POWER: In a vertical climb as drag becomes less of a factor as the airspeed drops, the aircraft's ability to pull itself forward (thrust) will help maintain a climb the best.  Pure physics.  It's why the La-7 and the Bf-109K-4 are climbing beasts.

LOW DRAG: In a vertical climb, as the P-51 and Spitfire climb, their low drag coefficients help reduce wasted energy in a climb with their moderately powerful engines assisting in the climb.

In practical terms, you could do the test two ways. One with the engine 100% with WEP on at 250MPH then point straight up . The second with aircraft at 250MPH then turn engines off and begin the climb.  The results will be completely different than the ones with engines on.

[Badboy]

As you see in F=ma, gravity isn't part of the equation. F=ma, even in a zero gravity enviornment. The force you speak of being calculated with that formula is inertia. Galileo was right.

It can be useful to demonstrate this by writing the equation of motion without the drag term so you get:

  f = ma

In this case the only force is gravity which is equal to mg so the equation becomes

  mg = ma

The mass term cancels from both sides of the equation giving:

    a = g

Which in effect tells us that if gravity is the only force, then the acceleration will always be the same and equal to g.  However the problem with that statement, is that the only force is not gravity, there are aerodynamic forces involved which can not be ignored when you are talking about aircraft.  Talking about a ball falling in a vacuum may make some sense in that context, but it makes no sense when talking about aircraft because you need an atmosphere to provide the forces that make flight possible. You can't then ignore air resistance no matter how convenient it might be.

The simple fact is that the only equations that make any sense include the drag term, and when you include the drag term the acceleration will be different when the mass is different as given by these two equations depending on if the object is going down or up   

 a = g - d/m

-a = g + d/m

These equations tell us that as long as the object is moving in air, which for aircraft they always will be, then the acceleration will depend on the mass.

They math and physics doesn't get much simpler than that.

Regards

Badboy

[Mister Fork]

It can be useful to demonstrate this by writing the equation of motion without the drag term so you get:

  f = ma

In this case the only force is gravity which is equal to mg so the equation becomes

  mg = ma

The mass term cancels from both sides of the equation giving:

    a = g

Which in effect tells us that if gravity is the only force, then the acceleration will always be the same and equal to g.  However the problem with that statement, is that the only force is not gravity, there are aerodynamic forces involved which can not be ignored when you are talking about aircraft.  Talking about a ball falling in a vacuum may make some sense in that context, but it makes no sense when talking about aircraft because you need an atmosphere to provide the forces that make flight possible. You can't then ignore air resistance no matter how convenient it might be.

The simple fact is that the only equations that make any sense include the drag term, and when you include the drag term the acceleration will be different when the mass is different as given by these two equations depending on if the object is going down or up    

 a = g - d/m

-a = g + d/m

These equations tell us that as long as the object is moving in air, which for aircraft they always will be, then the acceleration will depend on the mass.

They math and physics doesn't get much simpler than that.

Regards

Badboy

Unless we also factor in drag coefficient, these figures mean absolutely nothing.  As aircraft speed is measured in HUNDREDS of miles an hour, it has a significant impact on it's ability to maintain E and top speed.

[Badboy]

Unless we also factor in drag coefficient, these figures mean absolutely nothing. 

Don't forget thrust as well.

For aircraft the situation is complicated and without a complete model all you can do is generalize about how certain things influence the whole.

Fortunately we are lucky to have a complete flight model provided by HTC, so we can all be flight test engineers

Regards

Badboy

[FLOOB]

Exactly, air resistance affects how fast an object falls. Heavier objects need bigger parachutes because heavier objects have more inertia. Not because gravity makes heavier objects fall faster, but because heavier objects have more inertia and require more resistance to be slowed.

Mass has no affect on the acceleration of falling objects. Gravity accelerates all objects at the same rate regardless of mass. Embrace it. It's the law.

[FLS]

Exactly, air resistance affects how fast an object falls. Heavier objects need bigger parachutes because heavier objects have more inertia. Not because gravity makes heavier objects fall faster, but because heavier objects have more inertia and require more resistance to be slowed.

Mass has no affect on the acceleration of falling objects. Embrace it. It's the law.

I think you're confusing the rate of acceleration with the amount of force required to achieve that rate.

[FLOOB]

I am sure that I am not.

[Zoney]

The Gravitational attraction of Chuck Norris keeps the Earth and everything on it from floating away.

[Badboy]

Exactly, air resistance affects how fast an object falls.

Not just air resistance, the acceleration is also influenced by the mass as you can see in this equation of motion:

a = g - d/m

Here two objects with the same drag will accelerate differently depending on their mass.

Mass has no affect on the acceleration of falling objects.

No, that is only true when the object is falling in a vacuum, which is not a very helpful assumption when discussing aircraft because they need the atmosphere to fly. Once you accept your object is falling through the atmosphere your statement no longer holds and the acceleration is influenced by the mass of the object as explained.

Gravity accelerates all objects at the same rate regardless of mass. Embrace it. It's the law.

You need to relax that embrace and let it go, because you don't find many objects falling in vacuums. The only law you need here is Newton's law with all the force terms included, because you can't leave out air resistance and get meaningful results.

Regards

Badboy

[Mongoose]

Nope, with the same air resistance the one with the greater mass will fall more quickly.

Nope.  Wrong.  With the same air resistance both objects will fall at the same rate, regardless of their mass.  Galileo supposedly did an experiment by dropping two balls of differing weights from the top of the leaning tower of Pisa, and both balls hit at the same time.

The Apollo astronauts confirmed it on the moon.  One of the astronauts dropped a feather and a hammer at the same time, and they hit the surface at the same time. 

So, if you take wind resistance out of the equation, two objects of differing masses will fall at the same speed.  So if you have two objects with same air resistance, they will fall at the same speed, even if one is much heavier than the other.  The mass will not affect the dive rate, only the aerodynamic drag will. 

However, when you pull out of the dive, the object with more mass will have more momentum, and will go higher in a zoom climb.

This is the way the world works.  It has been tested, and proven.

[Badboy]

Nope.  Wrong.  With the same air resistance both objects will fall at the same rate, regardless of their mass.

No, that's not the way it works and I'm going to try to explain it differently for you, and confirm it with an experiment of my own.  

The Apollo astronauts confirmed it on the moon. One of the astronauts dropped a feather and a hammer at the same time, and they hit the surface at the same time.

Ok, that part is correct and to confirm that we can use Newton's second law like this:

Newton's second law is f = ma

On the moon the only force is gravity and the force due to gravity is given by mg so the equation becomes:

  mg = ma

Now you see that the mass cancels out on both sides leaving g = a

which means that the acceleration equals gravity and the mass makes no difference. That is what you are saying, and I agree that on the moon or in a vacuum that is true. However, you then extend that to motion in the atmosphere and you simply can't do that. When you take into account the additional forces, the acceleration then depends on mass as well.

So, if you take wind resistance out of the equation, two objects of differing masses will fall at the same speed.

Yes, that's exactly what happens on the moon, no atmosphere, no resistance, no difference in speed and we just used Newton's second law to confirm that.

So if you have two objects with same air resistance, they will fall at the same speed, even if one is much heavier than the other.

No, you can't extend your argument about what happens on the moon to what happens in atmospheric flight, that's a non sequitur. I will shortly prove that mathematically and confirm it experimentally.

The mass will not affect the dive rate, only the aerodynamic drag will.

Again, this is incorrect, both the drag and the mass affect the acceleration. In order to prove that first we can use Newton's second law to demonstrate the truth of it mathematically and then carry out an experiment to confirm it. The experiment was fun, I took some time between teaching today and a couple of my students volunteered to do the experiment so I could show it to you. It isn't as dramatic as dropping cannon balls from a tower, but the results are just as impressive.

Ok, firstly the math and again we will begin with Newton's second law, f=ma. However now the force is not just the force of gravity, there is also the air resistance which is represented with the letter d, so the force is now the gravity acting downwards and the resistance acting upwards so we have f = mg-d and Newtons law can now be written as:

   ma = mg - d

now if we divide both sides by m we get:

     a = g - d/m

You can read this as the acceleration equals gravity minus the drag divided by the mass. This makes it clear that the acceleration has an upper limit of g, and how much acceleration depends on the value of three things, g, d and m. You can also see that if you apply this equation to two objects that are identical in every way accept their mass, the heavier one will fall faster because d/m will be smaller, and it will reach a higher terminal velocity.

Now for the experiment.

After I read your message I devised an experiment to confirm the theory that was quick and easy to carry out.

Firstly I took two identical table tennis balls, and used a syringe to fill one with water.   



These two balls then had identical drag, the only difference being that one had greater mass.

According to you, they should both accelerate at the same rate and hit the ground together. According to Newton's second law the heavier one should accelerate faster and hit the ground first.

This is what happened when one of my students dropped the two table tennis balls at the same time, the ball filled with air is in his left hand and the one filled with water is in his right hand.

www.leonbadboysmith.com/video/Experiment.MOV

If you use the pause button to watch what happens you will see that the lighter ball in the left hand appears to be released a fraction of a second before the other and when it passes the dropper's chin it is slightly in the lead, but the heavier ball quickly overtakes the lighter ball and hits the ground with a lead of about four ball diameters. This experiment confirms the theory. In an atmosphere, mass does affect how fast things fall.

This is the way the world works.

Fortunately, it actually works in accordance with Newton's laws, if not those Apollo astronauts would never have made it to the moon in the first place.

Hope this helps...

Badboy
 

[Brooke]

Nope.  Wrong.  With the same air resistance both objects will fall at the same rate, regardless of their mass.  Galileo supposedly did an experiment by dropping two balls of differing weights from the top of the leaning tower of Pisa, and both balls hit at the same time.

That's because, for those spheres, D/m is small at the speeds experienced in a drop from the tower.  As pointed out, a = g - D/m.  When D/m is small compared to g, the equation of motion is approximately independent of m.

However, if d/m is not small compared to g, it matters.  Drop a feather and a bowling ball from the Tower of Pisa, and the bowling ball will hit the ground first.  That's because for the bowling ball g >> D/m and for the feather is about the same size as g.  This is an extreme example, but I'll give another one below.

The Apollo astronauts confirmed it on the moon.  One of the astronauts dropped a feather and a hammer at the same time, and they hit the surface at the same time. 

That's because there is no air on the moon and thus no d.  In this case ma = mg, thus a = g, and the equation of motion is independent of mass.

---- Case with More Detail ----

In more detail, D = 0.5 * rho * v^2 * A * C_D, where rho is air density, v is object velocity through the air, A is frontal area, and C_D is coefficient of drag.

Let's take the example of two sphere-like objects of exactly the same size but radically different masses:  an air balloon (filled with air, not helium, so we are not talking about buoyancy here) and the same balloon of the same size filled with water.  Intuitively, you know that dropping them from the Tower of Pisa, the air balloon will drift down slowly and that the water balloon will plummet down and hit first.  In terms of math, it is only because, for the water balloon, g >> D/m, but not for the air balloon.

Let's see what these numbers really are.  g = 9.8 m/s^2.  Let's take a balloon that is 10" diameter (like these http://www.amazon.com/Shipping-Free--germany-Advertising-Natural-Balloons/dp/B00809SWMQ/ref=sr_1_4?s=toys-and-games&srs=3017937011&ie=UTF8&qid=1389916292&sr=1-4&keywords=balloons ), so A = pi * (10 * 2.54 / 100 / 2)^2 = pi * 0.13^2 = 0.05 m^2.  The package of 100 of these is 0.36 kg, so each balloon weighs 0.0036 kg at most.  rho at sea level is 1.2 kg/m^3.  For a sphere, C_D is about 0.5.  Volume of the balloon is about 4/3 * pi * r^3 = 4/3 * 3.1415 * 0.13^3 = 0.0092 m^3.  Water density is 1000 kg/m^3, so that volume of water has a mass of 9.2 kg.

Thus, for the air balloon, D/m = 0.5 * rho * A * C_D / m * v^2 = 0.5 * 1.2 * 0.05 * 0.5 / 0.0036 * v^2 = 4.2 * v^2.  

For the water balloon, D/m = 0.5 * 1.2 * 0.05 * 0.5 / (0.0036 + 9.2) * v^2 = 0.0016 * v^2.  

D/m for the air-filled balloon is 2600 times higher than D/m for the water balloon.  At D/m of the air balloon is equal to g (i.e. terminal velocity) already at only 1.5 m/s.  It will accelerate to 1.5 m/s then drift down the rest of the way at 1.5 m/s.

For the water balloon, at 1.5 m/s, D/m is 0.0036 m/s^2 -- negligible compared to 9.8 m/s^2.  Terminal velocity of the water balloon is 78 m/s.  So, for the water balloon, D/m is insignificant in a drop test (except for very long drops, where v can get up near 78 m/s for a significant portion of the test).

This is all assuming I haven't made any math errors, of course.

Drop these two on the moon (where rho is almost zero), and they will drop the same.  Drop them in our sea level atmosphere, and you will get very different behavior.