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## Fun with Wind-Resistance (Part 2) – Optimal throwing angle

More fun with wind-resistance! (The cricket season starts for me tomorrow. Cracking.)

Last time, I showed a few trajectories of cricket balls (or baseballs) thrown in the presence of wind-resistance. I noted that I had chosen the angle of the throw in order to maximise the range of the throw. This optimal angle changes as the throw speed changes, as shown below.

The first thing to note in that the optimal throwing angle in the absence of wind-resistance is not 45 degrees, because the ball is released from 1.8m above the ground. (It would be 45 degrees if thrown from ground level). The angle is significantly less than 45 degrees at low speeds – maximum range requires a balance between vertical velocity (giving you more air-time) and horizontal velocity (giving you more range). The height of the thrower gives the ball extra air-time for free, so the thrower should use a flatter launch angle when throwing speed is small.

In the presence of wind-resistance, the optimal throwing angle drops below 45 degrees for very fast throws. The second, descending part of the balls trajectory will be slower and steeper than it would be in the absence of wind-resistance, so our thrower should opt for a flatter trajectory to take advantage of the higher velocity of the ball during its ascent.

In short, about 40 degrees should do it. Next time – will making the cricket ball heavier help?

Part One: Fun with Wind-Resistance

Part Two: Optimal throwing angle

Part Three: Optimal Mass

Part Four: Hitting at altitude

## Fun with Wind-Resistance (Part 1)

It’s finally happened. After a decade of dealing with frictionless slopes, massless strings, perfect vacuums and other spherical cows, I’m ready to complicate my model. What follows is a simple model for wind resistance, as outlined in University Physics by Young and Freedman. We’ll then have a look at the effect of air resistance on throwing a cricket ball (or baseball, if you must.)

In the absence of wind resistance, the equation of motion for a projectile is quite simple:

$a_x = 0$

$a_y = -g$

In the x-direction (horizontally), the ball moves with whatever horizontal velocity the thrower  gave it to start with. In the y-direction (vertically), the ball is pulled downwards, its vertical velocity changing at the constant rate of 9.8 m/s/s.

Wind resistance adds an extra force, one that pushes in the opposite direction to the way the ball is going. The magnitude of the force (for sufficiently large Reynolds number) is

$F_D = \frac{1}{2}\rho v^2 C_d A$

where $latex v$ is the speed of the ball, $\rho$ is the density of air $(1.2 kg/m^3)$, $A$ is the cross sectional area of the ball and $C_d$ is a dimensionless factor called the drag coefficient.

Because the drag force increases with velocity, a falling ball will accelerate until it reaches terminal velocity, where the drag force balances gravity. Thereafter, the ball falls with a constant velocity. The terminal velocity is given by:

$v_{t} = \sqrt{\frac{2mg}{\rho A C_d}}$

In practice, we use this formula in a different way. The terminal velocity is measurable, so we can use it to constrain the drag coefficient $C_d$. E.g. for a cricket ball, the terminal velocity is 123 km/h.

We now have all the pieces we need. The equation of motion is not solvable analytically, but is easily handled by any good numerical ODE solver. I’ll be using those of Matlab.

Let’s start with a few trajectories. I’m assuming that the thrower releases the ball from 1.8m.

## Got a cosmology question?

I’m thinking of setting my 3rd year cosmology students questions from the general public to see how well they’ve been listening, and to prepare them for unexpected cosmology questions at dinner parties. So I need all your questions about the universe!

Examples:

• What is the universe expanding into?
• Using powerful telescopes we can look ‘back’ at light from very early on in the universe. But how did we beat it ‘here’? It has been traveling at the speed of light and yet we are already here waiting for its arrival.
• If everything is expanding, how would we know about it, since all our rulers are twice as big as well?
• I heard an Australian guy won the Nobel prize for his work in cosmology. What did he actually do?
• Where did the big bang happen?
• What is redshift? How do we know what wavelength the light left the galaxy with?