Video: Pack 2 β€’ Paper 3 β€’ Question 15

Pack 2 β€’ Paper 3 β€’ Question 15

03:53

Video Transcript

A ball is thrown upwards. The graph shows the height of the ball in meters 𝑑 seconds after it was thrown. Work out an estimate for the gradient of the graph when 𝑑 is equal to 0.8. Show all your working.

To find the gradient of the curve, we first need to construct a tangent to the curve at the point given. On the π‘₯-axis of our graph, which represents time, 10 little squares represent one second. That means one little square must represent 0.1 seconds. 0.8 is therefore eight little squares, as shown.

A tangent is a straight line which touches a curve at one point only. Since we’re drawing this by eye, this method will give us an approximate answer for the gradient at that point. Once we have constructed a tangent to the curve that we are happy with, we can use the formula for the gradient of a straight line to calculate the gradient of the graph at this point.

Gradient is change in 𝑦 divided by change in π‘₯. You might also see this as 𝑦 two minus 𝑦 one divided by π‘₯ two minus π‘₯ one. We will need then to find two coordinates on our tangent. We can see that, on the 𝑦-axis, which represents the height of the ball, five little squares represent one meter. Dividing these both by five, we get that one square is equal to 0.2 meters. The two coordinates that lie on our tangent are therefore 0, 3.2 and 1.2, eight.

Remember, we said that the gradient was the change in 𝑦 divided by the change in π‘₯. Our 𝑦-coordinates are 3.2 and eight. So the change in 𝑦 is given by eight minus 3.2. Our π‘₯-coordinates are zero and 1.2. So the change in π‘₯ is given by 1.2 minus zero.

We could have actually calculated the gradient by subtracting these values the other way round. It doesn’t matter which direction you choose to subtract as long as you are consistent. 3.2 minus eight over zero minus 1.2 will give us the same answer as our first calculation. Eight minus 3.2 is 4.8. And 1.2 minus zero is just 1.2.

We could type these into our calculator. But if we were in a non-calculator paper, we can multiply both the numerator and the denominator by 10 to give us 48 over 12. 48 divided by 12 is four. The gradient of the graph when 𝑑 is equal to 0.8 is four.

Fully describe what your answer to part a represents. We found the gradient of the curve when 𝑑 is equal to 0.8. Remember, we said that gradient is change in 𝑦 divided by change in π‘₯. Looking at the context of our question, we’ve worked out the gradient as the change in height, since height is the 𝑦-axis, over a time given, since the time is the π‘₯-axis.

We could also consider the change in height as being a change in distance. So we’ve actually worked out distance divided by time. We know that speed is equal to distance over time. So we’ve actually worked out the speed of the ball 0.8 seconds after it was thrown.

Explain why your answer to part a is only an estimate. Remember, we drew our tangent by eye. It’s not possible to draw the tangent completely accurately. So this method will only give us an approximate answer or an estimate for the gradient at that point.

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