# Kick it

Algebra Level 5

A standard soccer goal is approximately 7.3 meters wide. 2 meters in front of it, and 0.5 meters to the right of its center, stands the goalie, poised, staring at you with a ferocious intensity.

The ball is at your feet, 10 meters out from the goal, and 2 meters to the right of its center. You have to kick it now, you have no choice, but since you are an excellent soccer player, you can of course kick it in any direction with perfect accuracy.

The ball will fly from you in a straight line, at constant velocity. The instant you kick the ball, the goalie will react perfectly - he will move, at a constant 10m/s, in the optimal path to intercept the ball before it can reach the goal; that is, if he can.

How fast do you need to kick the ball to get it past the goalie and score a goal? Let $$S$$ be the minimum speed, in meters per second, you need to kick the ball at to get it past the goalie and score that goal. Answer $$99S$$ rounded to the nearest integer.

Details:

• This may help. It's a simple diagram which shows the positions of all the things described in the above paragraph, on a to-scale graph.
• Treat all numbers specified in the above paragraphs as exact. Do not apply sig. fig. limitations to your calculations.
• Assume that the goalie, the ball, and the goal posts are all point masses with no thickness. The goalie can intercept the ball if and only if he can bring his point mass to exactly coincide with the ball's before the ball reaches the goal. Similarly, a goal is scored if and only if the ball's point mass manages to cross the 7.3 meter wide line segment that is the goal line without being intercepted by the goalie first.
• This problem is two dimensional. You cannot kick the ball off the ground, and the goalie cannot, and will not jump.
• If $$S$$ has no minimum, but rather an infimum, you should use that value for $$S$$ instead.
• Unless you thoroughly enjoy doing decimal math by hand over and over again, you're going to need a calculator to solve this. Please use a calculator for your own benefit!

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