Tweet
Maths Question. I'm seriously, you guys.
Easy for the Maths specialists here anyway, I expect.
I want to get a sense of the advantage gained by a goalkeeper in (Association) Football who, when facing a penalty kick, illegally jumps two yards forwards before the kick is taken.
This has become interesting because a certain goalie in the English Premier League keeps saving penalties by doing this, yet the officials don't seem to be able to spot the infringement and order the kick to be retaken. Obviously the officials are incompetent *****, but that is beside the point.
One way to express the advantage might be to estimate a percentage that represents how much 'smaller' the target area of the goal is for the penalty taker because of the 'narrowing of the angles' (as they say) by the goalkeeper being off his line. I'll give the dimensions and all the assumptions I can think of in a moment but a key thing is that the perfectly taken penalty is almost impossible to save by most goalkeepers as it will be tucked just inside the post, and out of reach of all but the tallest, most agile, far-leaping, and fastest-reactioned keepers. If the keeper starts off his line though, he has more chance of stopping the shot as his potential radius covers more of the surface area of the rectangle of the goal.
The goal is 24 ft wide and 8 ft high.
The penalty spot is 12 yards from the goal, in the centre.
Ball circumference is 28 inches.
Average velocity of a penalty kick is 24 yards/sec
Assume goalkeeper is 6 ft tall
Assume he can leap sideways 11 ft (tbh I just made that up)
The goalkeeper is not supposed to move before the ball is struck, but doing so seems to give a significant advantage, which is the purpose if this question. So assume the cheat-factor is that he moves 2 yards forward off his line before the kick is taken.
Can we ignore gravity and air-resistance on the ball and its weight to keep things simpler?
I don't really know how the advantage would be best expressed, so I'm open to improvements on the vague percentage thing I suggested above. Perhaps a formula where the encroachment, goalie height, kick velocity and especially leapability could be plugged in.
Rather than think about the probablity of a goalie saving a penalty (which I think over-complicates things), would it be sufficient to consider the goalie's available radius against the goal size in the two instances to get a sense of the advantage? If so then perhaps it becomes a trivial problem that even I could manage if I could remember any of my maths from all those hundreds of years ago. :0ldman:
Thanks in advance
Unrelated article on penalty biomechanics: http://www.usatoday.com/sports/2003-...lty-kick_x.htm
I want to get a sense of the advantage gained by a goalkeeper in (Association) Football who, when facing a penalty kick, illegally jumps two yards forwards before the kick is taken.
This has become interesting because a certain goalie in the English Premier League keeps saving penalties by doing this, yet the officials don't seem to be able to spot the infringement and order the kick to be retaken. Obviously the officials are incompetent *****, but that is beside the point.
One way to express the advantage might be to estimate a percentage that represents how much 'smaller' the target area of the goal is for the penalty taker because of the 'narrowing of the angles' (as they say) by the goalkeeper being off his line. I'll give the dimensions and all the assumptions I can think of in a moment but a key thing is that the perfectly taken penalty is almost impossible to save by most goalkeepers as it will be tucked just inside the post, and out of reach of all but the tallest, most agile, far-leaping, and fastest-reactioned keepers. If the keeper starts off his line though, he has more chance of stopping the shot as his potential radius covers more of the surface area of the rectangle of the goal.
The goal is 24 ft wide and 8 ft high.
The penalty spot is 12 yards from the goal, in the centre.
Ball circumference is 28 inches.
Average velocity of a penalty kick is 24 yards/sec
Assume goalkeeper is 6 ft tall
Assume he can leap sideways 11 ft (tbh I just made that up)
The goalkeeper is not supposed to move before the ball is struck, but doing so seems to give a significant advantage, which is the purpose if this question. So assume the cheat-factor is that he moves 2 yards forward off his line before the kick is taken.
Can we ignore gravity and air-resistance on the ball and its weight to keep things simpler?
I don't really know how the advantage would be best expressed, so I'm open to improvements on the vague percentage thing I suggested above. Perhaps a formula where the encroachment, goalie height, kick velocity and especially leapability could be plugged in.
Rather than think about the probablity of a goalie saving a penalty (which I think over-complicates things), would it be sufficient to consider the goalie's available radius against the goal size in the two instances to get a sense of the advantage? If so then perhaps it becomes a trivial problem that even I could manage if I could remember any of my maths from all those hundreds of years ago. :0ldman:
Thanks in advance
Unrelated article on penalty biomechanics: http://www.usatoday.com/sports/2003-...lty-kick_x.htm
Comment