What is the optimal position for a rugby kicker to convert a try?

[Karate Chop]

G'day guys, got this question out of Step-by-step calculs by J.G. graham for those of you who have the book.

I've had quite a bit of a play around with it but it doesn't seem to be going anywhere i end up with a few equations which have 3 variables, and i can't reduce it to two.

Okay here's the question.

In rugby, how far back from the tryline should the kicker take the ball to ave the best chance of converting the try?

Take the ball back until the angular width of the goalposts as seen by the kicker is a maximum. Let L be the width of the goalposts, let the try(touchdown thing) be scored a distance h (measured along the tryline EF) from the left hand goalpost (A) and let x be the distance the ball is take back perpendicular to the tryline, as shown in the diagram.

the field is 69 x 100 m and the width of the goalposts (L) is 5.6 m. Find the optimal position of the kicker inorder for him to have the largest angle to kick the ball through the posts.

 

[HallsofIvy]

the field is 69 x 100 m and the width of the goalposts (L) is 5.6 m. Find the optimal position of the kicker inorder for him to have the largest angle to kick the ball through the posts.

Yes, that's exactly what you want to do!

According to your picture, tan(angle BCD)= (h+l)/x so angle BCD= arctan((h+l)/x)
tan (angle ACD)= h/x so angle ACD= arctan(h/x). Therefore, angle BCA= angle BCD- angle ACD= arctan((h+l)/x)- h/x. h and l are constants, differentiate that with respect to x and set equal to 0.

 

[tacman]

The first step in solving this optimization problem is to define the variables and constraints. In this case, we have the distance from the left goalpost (h), the width of the goalposts (L), and the distance the ball is taken back (x). The constraint is that the field is 69 x 100 m and the width of the goalposts is 5.6 m.

Next, we need to find an equation for the angle of the goalposts as seen by the kicker. To do this, we can use the Pythagorean Theorem to find the hypotenuse of the right triangle formed by the goalposts and the ball. The hypotenuse will be the distance from the ball to the midpoint of the goalposts, which can be found by taking the average of the distance from the left goalpost (h) and the distance from the right goalpost (69-L-h).

Using this information, we can then use the inverse tangent function to find the angle of the goalposts as seen by the kicker. This equation would look like:

θ = arctan(5.6/2x + h - 69/2)

Now, we can take the derivative of this equation with respect to x to find the critical point where the angle is at a maximum. Setting the derivative equal to 0 and solving for x will give us the optimal position of the kicker.

However, as you mentioned in your response, this will result in a third variable, h, which we cannot eliminate. In this case, we may need to use a numerical method, such as trial and error, to find the optimal position of the kicker. This would involve plugging in different values for h and solving for x to see which value gives the largest angle.

Alternatively, we could also use a computer program or graphing calculator to plot the angle as a function of h and visually determine the maximum angle.

Overall, this is a challenging optimization question and may require some trial and error or the use of more advanced mathematical techniques to find the optimal solution. I hope this helps!