Math Is Fun Forum

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Tanθ>θ, for 0<θ<Pi/2

This proporsition is used for pointing out how to run to return a tennis ball.

Suppose the ball is coming to you right, would you move parallel along the baseline and hit it?
Or would you run an arc to meet it?

Imagine a ball trajectory in your mind. Because the ball travels much faster than you, let's ignore the time it takes from one point to a latter. Think the ball is everywhere on the trajectory. So you reach the trajectory, you reach the ball. Which way is shorter for you to reach it?

The second way, just an arc with start parallel to the baseline, ending up perpendicular to the ball trajectory.
It's always shorter than the tangent running.

And the difference is between a tangent and its angle. Think why.

Tanθ>θ, for 0<θ<Pi/2
It's true. Think why. Just imagine the plot of tan(x) and y=x for more clue.

Yes, the total distance the ball travels in the first way is larger than that in the latter way, indicating there is more time allowed before hitting in the first way. But the ball goes very fast, around 200km/s, compared to a low human running. So the arc provides more chance to hit the ball in time, reaching the trajectory sooner, without having to hit the ball earlier significantly.

This improved footwork is just a begining of revolutionary tennis.
For more, visit its site. www.revolutionarytennis.com

Last edited by George,Y (2007-05-30 14:20:12)

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X'(y-Xβ)=0

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Tanθ>θ, for 0<θ<Pi/2

Does the slight arc position your body better for hitting?
Also, what about going in a straight line close to perpendicular to the motion of the ball?

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igloo myrtilles fourmis

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Tanθ>θ, for 0<θ<Pi/2

Yes, the perpendicular distance is the shortest.
But there are pratical problems.
Your feet is hard to adjust to the very correct angle at the beginning.
And you may hit the ball to early, the ball is too tough.
Also after each hitting you are supposed to go back to the original level, otherwise you will get closer and closer to the net compulsively. This means perpendicular moving bears long road back.

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X'(y-Xβ)=0

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Tanθ>θ, for 0<θ<Pi/2

So it's nice to return the serve with the arc approach.

The coming back should be made after hitting. It might be a little clumsy to be frank.

But the most benefit is, by arc approach you face the ball straight with power. It's very important for corner returns. Suppose you've just run along the baseline, you have to turn inward before hitting.

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X'(y-Xβ)=0