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## Pythagorean theorem

# 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.