### Video Transcript

A ball is dropped vertically from a
height of 180 metres. It falls ๐ metres in ๐ก seconds,
where ๐ is directly proportional to the square of ๐ก. After two seconds, the ball has
fallen 20 metres. For how many more seconds will the
ball fall before it hits the ground?

So in this question, we have a ball
and itโs dropped vertically from a height of 180 metres. Iโve drawn that on a little diagram
here. And weโre also told that there is a
relationship between the distance fallen and the time in seconds, where ๐ is
directly proportional to the square of ๐ก. And Iโve written that down
here. And this sign here means
proportional. And when something is directly
proportional to something else, it means that it increases with it.

So we know that as the time
increases, well, the square of the time increases, then the distance will also
increase. However, this wonโt be much use to
us because we only need an equation if we want to find out any more information in
the question. And we can set one up using ๐
because we can say that ๐ is equal to ๐๐ก squared, where ๐ is called the
proportionality constant. And this allows us to have an
equation.

Now, with any question like this,
we have to follow the same steps. First of all, we set ourselves up
an equation. So weโve got ๐ equals ๐๐ก
squared. The next stage is to find ๐. And to do that, weโre gonna
substitute in some information weโve been given. Well, weโre told that after two
seconds, the ball has fallen 20 metres. So therefore, we know that when the
time is equal to two, the distance will be equal to 20. And we can substitute this in to ๐
equals ๐๐ก squared to help us find ๐.

So when we do this, we get 20,
because thatโs our distance, is equal to ๐ multiplied by two squared. And thatโs because two is our
time. So therefore, we have 20 is equal
to four ๐. And thatโs because two squared is
four. So therefore, to find ๐, what
weโre gonna do is divide each side of the equation by four. And when we do that, we get five is
equal to ๐. Or, we can say that ๐ is equal to
five. So great, weโve done the first
step. And weโve found ๐.

So now what we do is we substitute
our value of ๐ into our original equation. So weโve now got ๐ is equal to
five ๐ก squared. So this means that we can now use
this to find any value of ๐, given a value of ๐ก, or any value of ๐ก, given a value
of ๐. Well, what do we want to find in
this question? Well, letโs take a look. It says for how many more seconds
will the ball fall before it hits the ground.

Well, if weโre waiting for the ball
to hit the ground, then we know that ๐ is gonna be equal to 180 metres. Thatโs because we know that the
ball was dropped from a vertical height of 180 metres. So what weโre going to do is
substitute in ๐ equals 180 into ๐ equal to five ๐ก squared. And when we do that, we get 180 is
equal to five ๐ก squared. So therefore, the first step is to
divide each side of the equation by five. So then we get 36 is equal to ๐ก
squared. And thatโs because 36 goes into 180
five times.

And now, as weโre trying to find
๐ก, what we need to do is square root both sides of the equation. And when we do that, we get six
equal to ๐ก or we can say ๐ก is equal to six. We donโt have to worry about the
negative value because weโre talking about a time. So we only want the positive
value. So therefore, have we solved the
problem?

Weโve found out that the time is
six seconds for the ball to drop 180 metres, so to drop to the floor. Well, no because in the question
what we want to do is find out for how many more seconds the ball will fall before
it hits the ground. Well if we remember, we travelled
the first 20 metres in two seconds. And then the whole journey of the
ball, so from the start to hitting the floor, is going to be six seconds. So we need to find the difference
between the two.

So therefore, itโs gonna be six
minus two. And thatโs because thatโs the time
for the whole journey minus the time for the 20 metres which was two seconds which
is gonna give us four more seconds. So we can say that the ball is
gonna have to fall for four more seconds before it hits the ground.