Bending a Soccer Ball

Students from the University of Leicester have published a beautiful short research paper (pdf) on the physics of curving a soccer ball through the air.

It has been found that the amount a football bends depends linearly on the speed of the ball and the amount of spin.

— Sandhu et al., 2011: How to score a goal (pdf) in the University of Leicester’s Journal of Physics Special Topics

They derive the relationship from Bernoulli’s equation using some pretty straightforward algebra. The force (F) perpendicular to the ball’s motion that causes it to curl is:

F = 2 \pi R^3 \rho \omega v

and the distance the ball curls can be calculated from:

D = \frac{\pi R^3 \rho \omega}{ v m } x^2

where:

  • F = force perpendicular to the direction the ball is kicked
  • D = perpendicular distance the ball moves to the direction it is kicked (the amount of curl)
  • R = radius of the ball
  • ρ = density of the air
  • ω = angular velocity of the ball
  • v = velocity of the ball (in the direction it is kicked)
  • m = mass of the ball
  • x = distance traveled in the direction the ball is kicked

The paper itself is an excellent example of what a short, student research paper should look like. And there are number of neat followup projects that advanced, high-school, physics/calculus students could take on, such as: considering the vertical dimension — how much time it take for the ball to rise and fall over the wall; creating a model (VPython) of the motion of the ball; and adding in the slowing of the ball due to air friction.

↬ScienceDaily

The Hazards of Too Much Technology

New technology has a tendency to be used badly, but that does not mean it can’t be a powerful tool. Konstantin Kakaes argues that the increased use of technology is hurting science and math education.

A 2007 congressionally mandated study by the National Center for Educational Evaluation and Regional Assistance found that 16 of the best reading and mathematics learning software packages—selected by experts from 160 submissions—did not have a measurable effect on test scores.

— Kakaes (2012): Why Johnny Can’t Add Without a Calculator in Slate.

He makes some good points –a lot of technology is used employed simply because it’s “new technology” and not for what it can do– but I think he’s missing one fundamental aspect, probably because stuff is so new that we’re still figuring out how to use technology properly. The key missing aspect is that the increasing ubiquity of technology is changing who we are.

Technology is like an amplifier for our cognitive abilities –memorizing facts is less important because you can quickly look up the answers; how much time should you spend solving matricies if your can program your own matrix solver? –, and technology is becoming more closely integrated into who we are –we’re becoming inseparable from our smartphones (and it’s only a matter of time before they become implants).

A Review of Fractions: Based on Khan Academy Lessons

This is a basic review of working with fractions using lessons and practice sets from the Khan Academy.

1. Adding Fractions with a Common Denominator

The first topic — adding fractions –ought to be really easy for algebra students, but it allows them to become familiar with the Khan Academy website and doing the practice sets.

Now do the Practice Set.

OPTIONAL: Subtracting fractions with a common denominator works the same way. Students may do this practice set if they find it useful.

2. Adding Fractions with a Different Denominator

This is usually a helpful review.

The practice set.

3. Multiplying and Dividing Fractions

A good review that helps build up to working with radical numbers.

Multiplying fractions:

Do the multiplying fractions practice set.

Dividing Fractions:

Dividing fractions practice set.

4. Converting Fractions to Decimals

The last review is on how to convert fractions to decimals.

Now try the practice set for ordering numbers.

5. Next: Working with Square Roots