Video: AQA GCSE Mathematics Higher Tier Pack 2 โ€ข Paper 2 โ€ข Question 15

A ball is dropped vertically from a height of 180 metres. It falls ๐‘‘ metres in ๐‘ก seconds, where ๐‘‘ is directly proportional to the square of ๐‘ก. After 2 seconds, the ball has fallen 20 metres. For how many more seconds will the ball fall before it hits the ground?

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Video Transcript

A ball is dropped vertically from a height of 180 metres. It falls ๐‘‘ metres in ๐‘ก seconds, where ๐‘‘ is directly proportional to the square of ๐‘ก. 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 itโ€™s dropped vertically from a height of 180 metres. Iโ€™ve drawn that on a little diagram here. And weโ€™re 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 ๐‘ก. And Iโ€™ve 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 wonโ€™t 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 ๐‘˜๐‘ก 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 weโ€™ve got ๐‘‘ equals ๐‘˜๐‘ก squared. The next stage is to find ๐‘˜. And to do that, weโ€™re gonna substitute in some information weโ€™ve been given. Well, weโ€™re 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 ๐‘˜๐‘ก squared to help us find ๐‘˜.

So when we do this, we get 20, because thatโ€™s our distance, is equal to ๐‘˜ multiplied by two squared. And thatโ€™s because two is our time. So therefore, we have 20 is equal to four ๐‘˜. And thatโ€™s because two squared is four. So therefore, to find ๐‘˜, what weโ€™re 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, weโ€™ve done the first step. And weโ€™ve found ๐‘˜.

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

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

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

Weโ€™ve 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, itโ€™s gonna be six minus two. And thatโ€™s because thatโ€™s 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.

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