Does a Glider Complete Its Loop Starting at 60 m/s with 30 m/s² Acceleration?

[StudentTM]

Homework Statement

I have to write a C# program for an assignment for school. Programming is not a problem, but physics is :P So I am kindly asking you guys for helping me out with this.. Sorry for my bad english, I'm not a native speaker.

A glider pilot starts a back-loop with a plane, flying horizontally at a speed of 60 m/s. Maneuver is carried out so that he pulls the stick back and holds it in the same position througout the whole loop. The initial centripetal acceleration is 30 m/s2 in the direction vertically upwards.

Homework Equations

I must calculate the position, velocity and acceleration of the aircraft as a function of time during the interval from the beginning
implementation of acrobatics to the time in which the aircraft at a constant acceleration is outlining a full circle.
Does the aircraft come back to the original/initial position?
The origin is placed at the point at which the airplane moves from the horizontal position of the ribbon/loop.

I really hope I translated it so you can understand.

The Attempt at a Solution

I'm not really good at physics so I don't even know how to start :/

Acceleration due to gravity is 9.8 m/s2
Air resistance can be neglected, so that the only force acting on the plane is the force of gravity and the lift force, which is perpendicular to the direction of the current speed of the airplane.
Surrounding air is stationary relative to Earth and a plane (throughout the maneuver) doesn't rotate around its horizontal or vertical axis.

 

[voko]

Shall we assume that the pilot maintains constant airspeed throughout the maneuver? If not, I am afraid the problem is underspecified.

Anyway. You start with Newton's law that relates acceleration with force. I am sure you have heard about it.

 

[StudentTM]

I forgot to add a hint which was given, sorry:

From the initial centripetal acceleration reduced by the gravitational acceleration and velocity, we can calculate proportionality coefficient between the lift force and the square of the speed of the aircraft, which was assumed to be a constant.

But I just don't understand it :/
Newton's law? You mean F = m x a ?

Thank you for a reply

 

[StudentTM]

Due to the centripetal acceleration a plane starts to rise and the airplane due to the force of gravity loses speed
(Because the weight is no longer working at right angles to the direction of velocity). Due to the reduced speed decreases buoyancy (as it is proportional to the square of the speed), which affects the smaller centripetal acceleration. If the speed does not decrease, to the movement of the arc upward centripetal acceleration grow as the force of gravity is no longer directed opposite to the force of buoyancy.
The situations described are valid until the aircraft reaches the upper point of tangles, then the situation
turns.

 

[voko]

Yes, I mean that law.

And the hint is valuable, too. I suggest you refresh your memory on the lift force and its relationship with the square of the speed.

 

[StudentTM]

Voko, I still can't figure it out? Do you know how to solve this? Can you please help me with direct info? I must get this right till wednesdy and still have to do the programming.

 

[voko]

I cannot solve this for you, this is against the rules of the forum. I can only provide some help and guidance.

Newton's second law is a system of second-order differential equations, which you will have to integrate numerically. Do that sound like something familiar to you?

 

[StudentTM]

Well, this class is called 'Numerical methods' :) It is familiar, but I suck at it :/
Can you give me your email?

 

[HallsofIvy]

If the pilot is, indeed, flying in a perfect circle then his acceleration must be constant and always pointing toward the center of the circle. You say "initial centripetal acceleration is 30 m/s2 in the direction vertically upwards" so his acceleration is always 30 m/s^2 pointing toward the center of the circle.

Any circle can be given in parametric equations as $x= -R sin(\theta)$, $y= R cos(\theta)$ where, here, x is the horizontal distance right or left of the center of the circle and y is the vertical distance above or below the center of the circle. R is the radius of the circle and (x, y), when $\theta= 0$, is the starting position at the bottom of the loop. Assuming a constant angular speed, we can write $\theta= \omega t$ where $\omega$ is the angular velocity and t the time.

Differentiating, $dx/dt= -R\omega cos(\omega t)$ and $dy/dt= -R\omega sin(\omega t)$.
We are told that, at the beginning of the loop, when t= 0, the airplane has a horizontal speed of 60 m/ so we have $dx/dt(0)= -R\omega= 44$.

The net acceleration is the second derivative, $d^2x/dt^2= R\omega^2 sin(\omega t)$ and $d^y/dt^2= R\omega^2 cos(\omega t)$. We are told that, at the beginning of the loop, when t= 0, the airplane has an initial vertical acceleration of 30 m/s^2 so we must have $R\omega^2= 30$.

From those two equations, $R\omega= 60$ and $R\omega^2= 30$, you can solve for both R and $\omega$ and so find the parametric equations for the airplane's trajectory. Add R to the y component to move the origin to the bottom of the loop.

 

[StudentTM]

OK, thank you very much! I really appreciatte it!
I've got another theory, so we can compare:

Fact that this is a glider. And ignoring air drag really simplifies things. With no drag the principle of conservation of energy says that the glider will complete a closed loop, returning to its starting position and speed at the end of the loop.

For your computer program I suggest that you divide time into small increments - say 1/10 second, and determine the change in acceleration, velocity, and position for each increment of time. The equations you will need are:

Since lift force is proportional to the square of the velocity, so is centripetal acceleration. From a = v^2/r, this means that Kv^2 = v^2/r, and r is therefore a constant. Not what I would have expected.

3. For each increment of time you have a starting position, velocity, and acceleration, in both x and y coordinates. You need to treat velocity and acceleration as vectors (i.e. they have magnitudes in both the x and y directions) The ending acceleration position can be calculated from

p_2 = p_1 + v_1t + (1/2)a_1(delta_t)^2

where v_1 and a_1 are the values of initial velocity and acceleration. Other equations:

v_2 = v_1 - a_1 delta t

a_2 = a_1 + a_lift - a_gravity

where a_lift is v^2 x (30/60^2) in the direction perpendicular to the velocity vector, and a_gravity is -g in the vertical direction.

-------------------------------------

What do you say about what's written above?
I have to study and try to understand both theories. It's very difficult for me.
What does 'derivative' mean in english? Is this it -> (x2)' = 2x ?

 

[voko]

Well, this class is called 'Numerical methods' :) It is familiar, but I suck at it :/
Can you give me your email?

The purpose of this forum is not to help you get good marks no matter what. It is to help you learn.

That means you must be making an effort, and you must show your effort to get help here. If you are looking for other kinds of "assistance", you will have to look elsewhere.

 

[StudentTM]

Voko, I understand & agree with you. But what if I don't understand 'Newton's second law is a system of second-order differential equations, which you will have to integrate numerically'

 

[voko]

This is Newton's second law: $$ m \vec a = \vec F .$$ The little arrows indicate that the symbols represent vectors. In this particular case, we can assume that the aircraft moves in two dimensions only; let's call these two dimensions "distance" (left to right) and "altitude" (bottom up). The vectors are then pairs of numbers, and the law is really two equations: $$ m\ddot x = F_x ; $$ $$ m \ddot z = F_z, $$ where $x$ is the distance, $z$ is the altitude, the two dots mean "second derivative", and $F_x$ and $F_z$ are the distance and altitude components of the force acting on the aircraft. This is your system of second-order differential equations.

 

[StudentTM]

Thank you. And this is for a glider plane right?

 

[voko]

Yes, "aircraft" = "glider". You need to understand what $\vec F$ is in this case, and find $F_x$ and $F_z$.

 

[StudentTM]

We are trying to solve this with Euler's method..

I'll try to translate this correctly.. I'll mark XXX equations I'd have to fill in.In this case we have a plane with acceleration, which is at all times orthogonal to the movement of a plane.
If we would describe the movement in a horizontal plane (without the effect of a gravitational force) we would get a movement along a circural path.

1) a = a_0 x ( V_2^2 / V_0^2 )

2) a_0So we have;

V_0 ... initial speed
XXX1 ... initial position
XXX2 ... speed of the plane
XXX3 ... radius of the circle
XXX4 ... initial slope of the plane at a start

We should get two equations:

3) XXX5

4) XXX6

where XXX7 is defined as

5) XXX8Here XXX9 is the speed of the plane in x direction, XXX10 is the speed of the plane in y direction.

From this equations we get two diferential equations for the position in the x and y direction.

6) XXX11

and

7) XXX12Somehow we have to take into account also a quadratic dependence of the components of the acceleration, perpendicular to the direction of the movement of the speed:

a(v) + k v*v

that's why

a = a_0 x ( V_2^2 / V_0^2 )

Along this aerodynamic component there is also the acceleration due to gravity, which always points down.

Now, I don't know what XXX's are .. Anyone?Also, someone suggested this (also by Euler's method)

There are four equations, because you want to calculate velocity and position in both x- and y- coordinates. Using Euler for velocity, it works:

Vx_2 = Vx_1 +Ax t, where Ax = 39.8(V)(Vx_1)/60^2

Vy_2 = Vy_1 + Ay t, where Ay = 39.8(V)(Vy_1)/60^2 - 9.8

And then Euler can be used again for position:

x_2 = x_1 + (Vx_1)t
y_2 = y_1 + (Vy_1)t
Still trying to figure out those XXX's

 

[voko]

If you assume no gravity (and no drag), then both the velocity and acceleration remain constant, and the glider will be flying in a circle. You do not need to integrate anything in that case. The complete solution for this case has already been given by HallsofIvy.

If, however, gravity is present (and this is what the problem states), all of what you wrote before "Also, someone suggested this (also by Euler's method)" is meaningless. You need to proceed as I indicated in #15.

Now, what you wrote after "Also, someone suggested this (also by Euler's method)" makes a lot more sense. These are (almost) the equations you should obtain by following my earlier advice. Two things are wrong: the signs in the Ax and Ay terms, and the magic number 39.8 - where is it even coming from?

 

[StudentTM]

I think 39.8 is initial centripetal acceleration + gravitational acceleration.
The equation for the magnitude of A_L (meaning acceleration due to lift) is A_L = (30+g)(V_1/60)^2. Note that it includes g - this is because the amount of lift at the start must provide enough force to overcome gravity plus accelerate upwards at 30 m/s^2.

Does the following make sense? :

θ = ω * t ... angle between the position of the plane in the vertical going through the center of the loop

 

[voko]

I think 39.8 is initial centripetal acceleration + gravitational acceleration.
The equation for the magnitude of A_L (meaning acceleration due to lift) is A_L = (30+g)(V_1/60)^2. Note that it includes g - this is because the amount of lift at the start must provide enough force to overcome gravity plus accelerate upwards at 30 m/s^2.

OK, that seems correct. But the signs are still wrong. The force of drag is always against the velocity.

Does the following make sense? :

θ = ω * t ... angle between the position of the plane in the vertical going through the center of the loop

That above does make sense to me, even though I think it is wrong. The real question here is, does that make sense to you?

 

[StudentTM]

I try to understand but it's very hard for me. This above are equations of motion under the constrain of the centripetal force. But unfortunately, I think this is not going with Euler's method. My assignment is to program in C# with Euler's method. I think programming won't be so difficult when I get this physics theory (equations) right. But this is such a struggle :(

 

[voko]

You can use Euler's method to solve this problem. You just need to obtain the differential equations first. You were close to that in #16.

 

[StudentTM]

Yes, I'd have to manipulate the current vectors of position, velocity and acceleration. Finally this would lead me to get two differential equation for the second derivative of the two components of the position in time in dependence on the current position and velocity.

Any ideas how to do this?

 

[voko]

I indicated in posts #13 & #15, two days ago, what you need to do to get going. I do not see that you have made any attempt to make use of my advice.

 

[StudentTM]

I see, yes. But I don't know how to do that :/

So F = m * a ... a is centripental acceleration? If so, it's constantly pointed towards the center od the loop, right?
In #13, x and z are components of the acceleration?
How can it be distance and altitude .. If you multiply meter x meters you get square meter .. I'm confused.

 

[voko]

In #13, $\vec a$ is acceleration. It is the second derivative of position $\vec r$ with respect to time.

In this problem, acceleration is not constantly pointed to the center of the loop, and there is no single center of the loop.

Vector $\vec r = (x, z)$ is the position vector. It has two components: horizontal distance $x$ and vertical distance (altitude) $z$. $\ddot x = \dfrac {d^2x} {dt^2}$, second derivative of $x$ with respect to time; $\ddot z = \dfrac {d^2z} {dt^2}$, second derivative of $z$ with respect to time.

Vector $\vec F$ is the force acting on the glider. It is the sum of two forces: $\vec L$, which is the aerodynamic lift force, and $\vec P$, which is the weight of the glider. The magnitude and direction of $\vec P$ are constant; the magnitude and the direction of $\vec L$ both change depending on the velocity of the glider.

The direction of lift is always perpendicular to the direction of velocity; the magnitude of lift is proportional to the square of velocity, so $L = k v^2$. The proportionality coefficient $k$ you can determine from the initial condition, which specifies the acceleration and velocity at the beginning of the maneuver.

What I wrote above and in earlier posts is almost a complete solution of the physical part of the problem. All that remains to be done is project forces $\vec L$ and $\vec P$ onto the coordinate axes X and Z to obtains force components $F_x$ and $F_z$. This you will have to do yourself.

After that you need to convert the differential equations into the iterative formulae of Euler's method: $$ x_{n + 1} = x_n + ... \\ z_{n + 1} = z_n + ... \\ \dot x_{n + 1} = \dot x_n + ... \\ \dot z_{n + 1} = \dot z_n + ..., $$ which concludes the mathematical part of the problem.

 

[StudentTM]

Voko, thank you for your time and answer, I'll have a close look to what you wrote.

So here's the deal. I had to finish this till yesterday. I have to write the whole report and equation and how did I get those equations. Because I haven't been able to do the report on time, I sent to profesor a code qith equations I had and asked for more time. He said he don't have time to take a look at it, but will give me time till saturday to write the whole report and finish this right, so he sent me finished equations, which I must explain in the report. This is what he sent me (please Voko tell me if this is what I get if study your process)

 

[voko]

No, I do not think that you would get those equations if you followed the approach that I outlined above, at least not all of them. It also seems to me that those equations are mutually incompatible. For example, the second pair of the equations and the third pair, which both describe acceleration, differ in the constant term $-g$ in $a_y$.

 

[StudentTM]

What about this, what I sent to my professor (but he didn't check if it's right yet). These are simpler equations and a graph look right to me.

EQUATIONS:

// Acceleration
var Ax = -39.8 * V_t * Vy_1 / (60 * 60);
var Ay = 39.8 * V_t * Vx_1 / (60 * 60) - 9.8;

// Velocity
Vx_2 = Vx_1 + Ax * delta_t;
Vy_2 = Vy_1 + Ay * delta_t;

// Position
x_2 = x_1 + (Vx_1) * delta_t;
y_2 = y_1 + (Vy_1) * delta_t;RESULTS:

https://www.dropbox.com/s/4q1hgo0q99x0eag/loop.PNG

 

[voko]

If $V_t$ in the first pair means "speed" (magnitude of velocity), then everything looks good. And the shape of the trajectory also looks good.

I recommend that you compute, at each step, total mechanical energy (kinetic energy + potential energy). It should in theory stay constant because you ignore drag. But it will not be constant in your computations because of the error resulting from numeric integration. By choosing smaller time steps you can reduce the error, and the graph of total mechanical energy should give you an idea how significant the error is.

 

[StudentTM]

Aha, ok. So equation for that would be..

0,5 * m * V_t*V_t + m * g * x (height in this case is x) .. right?

And I take for example few positions, which give me V_t and x, for each of it. Total mechanical energy in each position must be almost the same (not exactly because of the error). Right?

So there are more ways of solving this through Euler method?

 

[voko]

Height in your case is y, not x. Good otherwise.

 

[StudentTM]

y of course :)

Thank you, I'll check that out

 

[StudentTM]

At the bottom of the loop where y = 0 and v = 60 m/s the KE is (1/2)mv^2 = 1800m J. At the top of the loop I have v=19.36m/s and y= 166.061, for a total energy of (1/2)mv^2 +mgh = 1814.8m J. I think the error is due to the way that the Euler method assumes for each incremental step that acceleration is constant in both x and y directions - it ignores the fact that the plane's angle has changed which would have the effect of slowing its forward velocity.
Should I include the angle somehow with this simplified equations?

 

[voko]

You have the error because Euler's method approximates a continuous function with a piecewise linear curve. This should have been explained to you in your "numerical methods" course. You cannot eliminate this error, but you can make it smaller by reducing your time step. Another source of error is the finite precision of computer arithmetic. This you cannot reduce at all (without taking some extreme measures, that go far beyond your little assignment).

Keeping track of total energy let's you estimate the error; your error is about 1%, which is not too bad, especially if you recall that you ignore drag completely, which will have a much greater effect on energy and other variables.

 

[StudentTM]

I understand. Thank you Voko!

I'll now start finishing this program and start writing report.
I might have a questions or two for the process of evolving those equations.
I'll post the finished assignment/program here. Thanks to everyone who helped and participate in discussion.