Current time:0:00Total duration:12:05
0 energy points
Studying for a test? Prepare with these 4 lessons on Equations and geometry.
See 4 lessons

Thiago asks: How much time does a goalkeeper have to react to a penalty kick?

Video transcript
THIAGO SILVA: [SPEAKING FOREIGN LANGUAGE] SALMAN KHAN: Great question, Tiago. And to understand it, let's look at the dimensions of a penalty kick. The kick itself is from 12 yards away from the goal or 36 feet. The goal itself is 24 feet wide or 8 yards wide. And the goal is 8 feet tall. And so let's think about a few other dimensions that might not be as obvious. Let's try to figure out the distance from the ball where it's kicked to the bottom right of the goal. And this, obviously, is going to be the same distance as to the bottom left of the goal. And I encourage anyone watching this to pause the video and think about that right now. Well, the way I've drawn it, you see that this is actually a right triangle. And so we could use the Pythagorean theorem in order to figure out what is this distance right over here. And you might say, well, wait. How do we figure that out? Well, we already know that this length of the triangle is 36 feet. And we know that this base right over here is 1/2 of the width of the goal. So this is going to be 12 feet right over there. So the Pythagorean theorem tells us that this distance right over here is going to be the square root of the square of the sum of the squares of the other side. So it's going to be the square root of 12 squared plus 36 squared. And let's get our calculator out and try to answer it and figure out what that is. So that's going to be the square root of 12 squared plus 36 feet squared-- oops, not 33-- 36 feet squared is equal to 37.9. So let's just say-- well, let's just use that number right now. So 37 point-- we could even say 0.95, almost 38 feet. So this is approximately equal to 37.95 feet. And that's going to be the same as this distance right over here. Now, let's figure out the distance, an even further distance, the distance to the top right, which is also going to be the same thing as the distance to the top left. And I'd encourage, once again, people to pause the video and try to think about that on their own. Well, let's draw another right triangle. And this one might not be as obvious. But if I draw the straight line distance of the ball to, in this case, the top right of the goal, I have now constructed another right triangle. Notice this is a 90 degree angle. One side is 37.95 feet. The other side is eight feet tall. And so this distance right over here is going to be the square root of 37.95 squared plus 8 feet squared. So let's figure out what that is, get the calculator out. So I can square the last entry on my calculator just by typing in this-- that just means take the last answer, square it-- and then add that to 8 squared, which we know is 64. And now we want to take the square root of that. So take the square root of 1,504, gets us to 38, we'll just say, roughly 38.8 feet, or let's say 0.78 feet. So this is approximately equal to 38.78 feet. Now the next thing I want to think about-- and I think this will be what we focus on-- to figure out how much time does the goalie have to get here, because one could argue that this is the hardest to get to, that the goalie, they have to travel the furthest. And they have to dive for this right over here. So let's think about the distance from this point to this point here. And then we can think about how much the goalie actually has to move because they have some height. And their hands, they can stick up in the air. And this one, once again, is a fairly straightforward Pythagorean theorem problem. You have a right triangle here. You could also see it on this side. It's a little easier. You have a right triangle here. We know that this is 12 feet. And that this right over here is 8 feet. So we know that this distance right over here is going to be the square root of 12 feet squared, which is 144, plus 8 feet squared, which is 64. So let's figure out what that is. So that's going to be the square root of 144 plus 64 is equal to 14.42 feet. It's equal to 14.42 feet. Now, we assume that the goalie isn't traveling all the way from here to all the way there. The goalie has some height. And he or she could stick their hands up in the air. So we could imagine a goalie stretched out like this, trying to dive for that ball. And so the actual distance that they have to travel is from the tip of their reach to that corner right over there. So if we assume that the entire, if a goalie stretched out, is, let's say, 7 and 1/2 feet completely stretched out-- so this distance, if this distance fully stretched out is 7.5 feet and they're trying to get 14.42 feet away-- and I could maybe start rounding down a little bit rougher numbers to, say, 14.4 feet-- then they need to travel about 6.9 feet. So they need to travel about 6.9 feet. So for this top right kick or this top left kick, the ball is going to travel 38, almost 39 feet, 38.8, 38.78 feet. And the goalie has to travel-- the goaltender has to travel 6.9 feet. Now that we know the distances that the ball has to travel and that the goalkeeper has to travel, we can now think about the time in which it's going to happen. And to do that, we're going to have make assumptions about their speed. So I did some research on the internet. And it looks like a penalty kick can go-- a fast penalty kick can be around 60 miles per hour, although it does look like there are documented cases of 80 miles per hour or even higher than that. But let's just say 60 miles per hour for a fast penalty kick. So this is the kick speed or the ball speed. And let's assume that this person can jump at 15 miles per hour, which is actually a pretty good speed from a standstill. So it actually might be a little bit aggressive. So jump speed-- I'll write it here-- jump speed of the goalkeeper. Let's write that as 15 miles per hour. And so to make sense of it, because everything else we've done in feet, let's convert these each into feet. So 60 miles per hour. If I want to convert it to feet, we just have to remind ourselves that 60 miles is the equivalent to 60 times 5,280 feet. Each mile is 5,280 feet. So this would give me the total number of feet in an hour. But we don't want feet per hour. We want feet per seconds. So this is how far you would go in feet in an hour. To figure it out in seconds, you would want to divide by 3,600 because there are 3,600 seconds in an hour. So this gets us to 88 feet per second, 88 feet per second for the ball. And now let's do the same thing for the goalkeeper. So 15 times 5,280-- so this is the feet traveled in an hour. But we want it in a second. So we're going to divide it by 3,600, gets us to 22 feet per second. So this is equal to 22 feet in a second, 22 feet per second. So now we can use these speeds to figure out how long will it take the ball to go from this point all the way to the top right corner. Well, we'll just have to remind ourselves that distance is equal to speed times time. Or if we want the time, we just have to take the distance an divide it by speed. So the time for the ball-- so ball time-- is going to be equal to 38 point-- let's just go with 38.8 feet-- had to make a lot of rough assumptions here anyway-- is going to be equal to 38.8 feet divided by 88 feet per second, which is equal to-- so 38.8 divided by 88 gets us 0.44 seconds. So let's write that. So that's 0.44 seconds or 44/100 of a second, a little under half of a second for this ball to reach there. Obviously, if the ball was going even faster, it would take even less time. If it was going slower, it would take a little more time. Now let's think about how far it would take for this person to travel the 6.9 feet. So the goalkeeper time, goalie time is equal to the 6.9 feet. We're assuming he's kind of already in this position, kind of already starting to stretch out. Or he stretches out while he's in the air when he launches himself. So it's going to be-- and, obviously, I'm making a lot of rough assumptions here-- 6.9 feet divided by 22 feet per second. So that gets us 6.9 divided by 22 is equal to 0.31-- I'll just round there-- is equal to 0.31 seconds. So just based on what we saw, the ball's going to take 44/100 of a second to get there. The goalkeeper, if we assume the 15 miles per hour, is going to take 31/100 of a second to get there. And so they only have the difference to make the decision where to jump and, even frankly, to start their jump, getting into the jumping position, to kind of scrunch up and jump a little bit. So the difference between these two things is only-- let me write this in a new color-- this is 13/100 of a second to make this decision. And that's why, frankly, penalty kicks are successful so frequently. Most people's reaction time-- and even professional athletes does not-- professional athletes get close to this in terms of reaction time. I did a little bit of research on the internet. Most other people's reaction time is nowhere near this low. It's often double this or higher. So even if they make the exact right decision and even if they're able to launch themselves up at 15 miles per hour, they have little over a tenth of a second to make that decision. Now once again, I want to emphasize, this was given all of these assumptions that I made. You might want to lower or higher this assumption of how fast they can jump. You might want to increase or decrease the assumption about how fast the ball is going. And you could also think about different points on the goal to see which one, based on your assumptions, might require different reaction times.