physics question about pull ups

I was trying to work out the expected height gain from a pull up
Experienced glider pilots say you will get a better pull up with a
heavier glider / water etc.
But I can't see this from my (probably incomplete) equations:

total energy = potential energy + kinetic energy

total energy before pull up = total energy after pull up

m * g * h0 + m * pow(v0, 2) * 0.5 == m * g * h1 + m * pow(v1, 2) * 0.5

with h0 v0 being height and speed before pull up
and h1 v1 being height and speed after pull up

mass cancels out of this equation

I think I need to include momentum in there somehow?

Hi John,

You are correct. The physics equations show that you will get the same
height regardless of the weight of the glider.

However, I think it is true that a heavier glider will have a slightly
higher pull-up. I don't think the difference is very much though. Both
gliders will have similar frictional losses and losses due to inefficiencies
during the pull-up.

Paul Remde

"John Rivers" wrote in message
...
I was trying to work out the expected height gain from a pull up
Experienced glider pilots say you will get a better pull up with a
heavier glider / water etc.
But I can't see this from my (probably incomplete) equations:

total energy = potential energy + kinetic energy

total energy before pull up = total energy after pull up

m * g * h0 + m * pow(v0, 2) * 0.5 == m * g * h1 + m * pow(v1, 2) * 0.5

with h0 v0 being height and speed before pull up
and h1 v1 being height and speed after pull up

mass cancels out of this equation

I think I need to include momentum in there somehow?

On Apr 20, 5:06*am, John Rivers wrote:
I was trying to work out the expected height gain from a pull up
Experienced glider pilots say you will get a better pull up with a
heavier glider / water etc.
But I can't see this from my (probably incomplete) equations:

total energy = potential energy + kinetic energy

total energy before pull up = total energy after pull up

m * g * h0 + m * pow(v0, 2) * 0.5 == m * g * h1 + m * pow(v1, 2) * 0.5

with h0 v0 being height and speed before pull up
and h1 v1 being height and speed after pull up

mass cancels out of this equation

I think I need to include momentum in there somehow?

You have included momentum :-)

I think the answer is in where on the L/D curve the glider is flying
during the pullup. And how close you can get to the optimal flight
path.

Your formula is correct but incomplete. It does not account for the
energy lost due to drag. Also, v1 (assuming it is stall speed) will
have some dependence on mass. However these are higher order effects;
in the first approximation you are correct.

On Apr 20, 1:06*am, John Rivers wrote:
I was trying to work out the expected height gain from a pull up
Experienced glider pilots say you will get a better pull up with a
heavier glider / water etc.
But I can't see this from my (probably incomplete) equations:

total energy = potential energy + kinetic energy

total energy before pull up = total energy after pull up

m * g * h0 + m * pow(v0, 2) * 0.5 == m * g * h1 + m * pow(v1, 2) * 0.5

with h0 v0 being height and speed before pull up
and h1 v1 being height and speed after pull up

mass cancels out of this equation

I think I need to include momentum in there somehow?

On Apr 20, 5:06*am, John Rivers wrote:
I was trying to work out the expected height gain from a pull up
Experienced glider pilots say you will get a better pull up with a
heavier glider / water etc.
But I can't see this from my (probably incomplete) equations:

total energy = potential energy + kinetic energy

total energy before pull up = total energy after pull up

m * g * h0 + m * pow(v0, 2) * 0.5 == m * g * h1 + m * pow(v1, 2) * 0.5

with h0 v0 being height and speed before pull up
and h1 v1 being height and speed after pull up

mass cancels out of this equation

I think I need to include momentum in there somehow?

You've also forgotten what the initial speeds are. When you are
flying with a heavier wing loading you are flying faster before the
pullup than you are with a lighter wing loading. Therefore, you'll
end up higher.

-- Matt

mattm wrote:
On Apr 20, 5:06 am, John Rivers wrote:
I was trying to work out the expected height gain from a pull up
Experienced glider pilots say you will get a better pull up with a
heavier glider / water etc.
But I can't see this from my (probably incomplete) equations:

total energy = potential energy + kinetic energy

total energy before pull up = total energy after pull up

m * g * h0 + m * pow(v0, 2) * 0.5 == m * g * h1 + m * pow(v1, 2) * 0.5

with h0 v0 being height and speed before pull up
and h1 v1 being height and speed after pull up

mass cancels out of this equation

I think I need to include momentum in there somehow?

You've also forgotten what the initial speeds are. When you are
flying with a heavier wing loading you are flying faster before the
pullup than you are with a lighter wing loading. Therefore, you'll
end up higher.

-- Matt

I think that this is the best brief answer too...


Brian W

On 20 Apr, 18:37, brian whatcott wrote:
mattm wrote:

You've also forgotten what the initial speeds are. *When you are
flying with a heavier wing loading you are flying faster before the
pullup than you are with a lighter wing loading. *Therefore, you'll
end up higher.

-- Matt

I think that this is the best brief answer too...

Brian W

I suspect that an least equal factor is the one that Toad mentioned:
for a given airspeed the heavier glider will be flying at a flatter LD
and will have less energy losses than the lighter one in being rotated
to the same climb angle.

John

Interesting problem... Seems like you've defined the concept of
energy height. Energy is height is just how you have described
already...
E= mgh + 0.5mv^2
OR
E = wh + (w / 2g)v^2

The effect of drag on height recovery isn't too bad, but is enough to
matter. In general the shallower the climb and the bigger the speed
change the greater effect drag will have overall.
As an approximation, a glider initiating a 30deg climb from 120kts to
40kts would only take about 8sec (without drag decel, only gravity).
Assuming an average L/D of 24:1 over the entire maneuver the glider
would loose only about 5.6 feet of altitude per second due to drag. So
that means about 45 feet of altitude would be eroded due to drag. Of
course that's just an approximation. I'm sure there's a more correct
way...

The only way I could think to explain the extra height gain with a
heavier glider of the same model is the relationship of L/D vs speed.
For most gliders at a given speed their L/D will be higher with more
weight. Looking at polars you'll see, on average, the ballasted
glider will hold a better L/D at the same speed almost all the way
back to thermalling speed. The difference is small but is enough to
matter. On top of that the heavier glider at the same speed has more
energy height available due to its mass...
www.valleysoaring.net/pk/x-c/polar24c.jpg

One thing I wonder is whether a heavier glider initiating a 1.5g climb
would loose more energy through induced drag than a lighter glider,
assuming the same speed. Since a heaver glider would require more
force (lift) to induce a 1.5g climb then I'd think that would require
a greater angle of attack change thus more induced drag.

wrote:
The effect of drag on height recovery isn't too bad, but is enough to
matter.

In a low-performance glider the drag can be extremely significant. In,
say, a K8 or (I'd guess) an I-26, the height gain is very small in
comparison with 40:1 glass.

A pilot flying at the UK Juniors a few years ago described a racing
finish in a K8, producing no more than a 200 ft climb from a 90kt
pull-up. He said that a K8 in this mode was the ultimate efficient
machine "for converting height into noise".

On Apr 20, 1:41*pm, Chris Reed wrote:
wrote:
The effect of drag on height recovery isn't too bad, but is enough to
matter. *

In a low-performance glider the drag can be extremely significant. In,
say, a K8 or (I'd guess) an I-26, the height gain is very small in
comparison with 40:1 glass.

A pilot flying at the UK Juniors a few years ago described a racing
finish in a K8, producing no more than a 200 ft climb from a 90kt
pull-up. He said that a K8 in this mode was the ultimate efficient
machine "for converting height into noise".

If you've not seen this site you're missing a lot of good information
on the aerodynamics of flight, including what John calls the "roller-
coaster" effect. http://www.av8n.com/how

It's written for power flight but most applies to gliders as well.