Cartoon: Austerity Wedding Plans

Next year's royal wedding between Prince William and Kate Middleton is being dubbed the 'austerity wedding' by the media. The Times' cartoonist Peter Brookes uses this idea as the basis for this cartoon.

The  Duke of Edinburgh and  Prince Charles are standing in the courtyard outside Buckingham Palace. The Chancellor of the Exchequer  George Osborne is waving a magic wand over a pumpkin and several white mice. The Duke remarks: "Is this the best you can bloody come up with, Osborne?"

EXPLANATION
In the story of  Cinderella, the pumpkin and white mice are magically transformed into a coach and horses by the Fairy Godmother. Since the government doesn't have any money to spare to pay for a lavish royal wedding due to the financial crisis, George Osborne is reduced to trying the same trick. Prince Philip (aka The Duke of Edinburgh) is not impressed!

VOCABULARY
If you come up with a plan or idea, you think of it and suggest it. 

The Curving Soccer Ball

You can call it football if it makes you happy. Anyway, this is a popular story going around. The physics of the magic curving soccer kick. Here are two ends of the spectrum.

Let me select one tiny part of the paper to show you: (they used pictures for some of the variables, so some of this might not appear exactly as the author intended - but you will get the idea):

"The motion of the sphere of mass M is described in the Serret-Frenet coordinate system introduced in figure 2. We first focus on the direction . The Reynolds number Re = ρU0 R/η is of the order of 104, which implies a drag F1/2ρU2πR2·CD, with CD0.4 [28]. The equation of motion along thus is written as"

They lost me at "Serret-Frenet" coordinate system. So, this doesn't appear to be consumable for the more general audiences.

So, you kick a ball. What forces act on the ball? Well, the easy thing is to say "gravity and stuff that touches the ball". In this case, the only thing the ball touches is the air. The air does indeed exert a force on the ball. The force the air exerts on the ball is ultimately due to collisions with the air particles and the ball. If the ball is spinning and non-smooth, there can be complex interactions. For this case, I will break this air force into two components.

• Air drag. If you have read this blog, you should be familiar with this model of air drag that says the force is proportional to the magnitude of the velocity squared and some other stuff (density of air, cross sectional area, and shape of the object).
• Magnus force. This is the force exerted on a moving and spinning object in a fluid or gas.

There is also the gravitational force. But, let me just look at the ball from the top view. The key point of all of this is that if there were no spin effect or air drag, the ball would just move in a nice parabola. From the top, this would look like a straight and constant speed trajectory. If you exert a force perpendicular to the direction of motion, the ball will turn. If you exert a force in the opposite direction of the motion, the ball will slow down. These two things together make the ball do what it does.

Here is a force diagram of the ball as seen from the top (so you don't see the gravitational force):

Why does this spinning cause a sideways force? Well, the idea is that the rough surface of the ball moves air near its surface. This means that on one side of the ball, the air is moving faster than the other side. On the faster moving air side, the air is moving more in a direction parallel to the motion of the ball. This means that an air particle is less likely to collide on the side of the ball and push it that way. The result is that there is more collisions on the slower side of the ball.

Here is the model that is commonly used for the air drag force:

Where the v-hat is a unit vector in the direction of the velocity of the ball. This along with the negative sign means that the air drag force is in the opposite direction as the velocity.

The magnus force can be written as:

S is some constant for the air resistance of the ball (a basketball and a soccer ball would have different values). The vector ω is the vector representing the angular velocity of the ball. For the diagram shown above, the vector ω would be perpendicular to the plane of the computer screen and coming out of the computer screen. The mangus force is related to the cross product of ω and the velocity. (here are some cross product tips).

Why don't you always notice these forces? If the speed is slow and the mass is large, then the air drag and magnus forces will be small compared to the gravitational force. The motion for these cases will be dominated by the gravitational interaction. But with a high speed kick from a soccer ball (that has a relatively low mass) with a high angular spin, the effects can be noticed.

Let me model a high speed soccer ball in vpython. The original research paper gives some nice parameters that I will need for a soccer ball.

• Radius = 0.105 meters
• density = 74 times the density of air (if I understand the table correctly)
• S = 0.21 - I am pretty sure the S in this paper is the same S in the magnus force described above. - forget this S

After playing around (and finding that third article) I am pretty sure the S above is not the same S as in the wikipedia page. The physicsworld article gives the following useful info:

• Ball speed = 25-30 m/s
• angular velocity = 8 - 10 rev/sec
• Lift force (magnus force) of about 3.5 N
• horizontal ball deviation of about 4 meters
• ball mass of 410-450 grams (which agrees with my previous density)
• ball acceleration of about 8 m/s² - not sure if this is just the linear acceleration or the total magnitude of the acceleration and at the beginning or average?

If I assume the magnus force is S times the cross product of the angular and linear velocity, I can work backwards to find S (from the physicsworld data) in the case that the velocity and angular velocity are perpendicular.

Now for some python (here is my sloppy code -

magnus_force.py). I will make one assumption - the angular velocity of the ball is constant (which obviously will not be true).

Here is what I get for the trajectory of the ball (as seen from above).

That is more than 4 meters deflection - but maybe they are assuming you aim to the left a little or something.

How about a plot of the total acceleration (magnitude) as a function of time.

This gives an acceleration of around 8 m/s² around the end of the motion. Maybe this is what the physicsworld author meant. Oh well, that is enough for this. I know there is one problem. I assumed a constant coefficient of drag, but it seems that this might not be true.

cool stuffs with physics!!

What is going on here and what does this have to do with g? Well, it doesn't really have anything to do with g, but it is a cool demo (meaning this demo would still work on the moon). The basic idea is that you aim a projectile launcher straight at a hanging target. When the ball is launched, the target drops and the ball collides with the target in mid-air.

Why does this work? Let me start with a diagram.

I guess it is clear, but the vector v₀ is the initial velocity of the ball aimed at an angle θ at a target that is a distance s horizontally away and a height h above the launcher. But why do they hit? Let me start with the ball. (here is a quick projectile motion review) The ball's motion can be broken into an x-direction motion and a y-direction motion with the following initial velocities:

Two quick notes: I am using v₀ to represent the magnitude of the initial launch speed. Also, the x-velocity does not change (no forces in the x-direction) so that I can just call it v_xinstead of v_x0 (of course, the y-velocity does change). Using those velocities, I can get position-time functions for both directions.

There is the "g"! Also, I am assuming the ball launches at time t= 0 seconds from a position x₀and y₀. Now, what about the target? The target also starts falling at t = 0 seconds, but its initial position is (x₀ + s,y₀ + h) as seen in the diagram. There is no x-direction motion for the target, so I can just write:

I am calling y_m is the y position of the target as a function of time. The target's initial velocity is zero. First question: what time will the ball be at the x-coordinate of the target? Going back to the function for the x-position of the ball,

And what will be the vertical position of the ball at this same time?

What about the target? Where will the target be at this same time?

Notice that the only difference between the y position of the target and the ball is that the ball has a "s tan(θ)" where the target just has an h term. Oh but wait. Look back at the original diagram and you will see that:

What does this mean? This means that at the time when the ball reaches the x-position of the target the ball and the target have the same y-position. If two things are at the same place at the same time, they hit. Boom.

Notice one cool thing - the value of the magnitude of the initial velocity does not matter (unless the ball hits the floor before it gets to the target). The only important thing is that the ball is aimed at the target. This means that θ for the velocity vector is the same θ for the geometrical set up of the target.

Wait! There is more! How about a nice python-generated graph? This is pretty simple to set up really. In python, I will model the shot ball and the falling target.

That is with a launch speed of 6.5 m/s. What if I increase the speed to 8 m/s?

You can't really see that the two paths cross at the same time. Let me draw fewer points.

There, that is with a time step of 0.1 seconds. The points are numbered and you can see that the two points labeled 4 are right next to each other.

That is it for g-day. I will have bigger plans for next year.

Cool ELT Sites

Check out the British Council's Learning English website. There are many activities that can be adapted and used by teachers in their classrooms. For older learners, perhaps we can try stretching their imagination with these podcasts based on stories and poems. They come with activities you can download or do online.Do check it out!

NEW LITERATURE COMPONENT: FORM 5 NOVEL 2011!!

It's been almost couple weeks since I'm away from school. Hence, it is no surprise that I only knew about the latest novels for Form Five English Literature Component about a few days ago. You might have heard about this from other blogs. Nonetheless, I'd like to share the information in this blog as well. Thanks Teacher Sab for the info :) I'm looking forward to reading the three books and posting my reviews on them soon. Check it out!

-The Curse by Lee Su Ann

(will be read by Pahang,Terengganu, Johor,Sabah,Sarawak and Labuan)

-Step by Wicked Step by Ann Fine

(will be read by Selangor,Kuala Lumpur,Putrajaya, Negeri Sembilan and Malacca)

-Catch Us If You Can by Catherine McPhail
(will be read by Perlis,Kelantan,Kedah, Penang and Perak)