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Projectiles problem

Hi, my friend and I were solving question 7:

https://ibb.co/5K26yLv

We have succeeded in getting through the problem.

However one thing is bugging us, in 7iii) we are asked to use the discriminant to find the greatest distance that the golf ball can clear the tree.

To do this it felt instinctive to us to set the discriminant equal to 0 and solve. This gave us a y value of 27.5, leading to the correct answer of 22.5

The thing that is troubling us is the following: why is it that the discriminant being =0 leads to the greatest value of y?

The discriminant tells us that there is only one solution, but we can't make the link as to why this leads to the greatest value of y.

An explanation would be much appreciated by both of us. Thanks in advance.
Original post by jc768
Hi, my friend and I were solving question 7:

https://ibb.co/5K26yLv

We have succeeded in getting through the problem.

However one thing is bugging us, in 7iii) we are asked to use the discriminant to find the greatest distance that the golf ball can clear the tree.

To do this it felt instinctive to us to set the discriminant equal to 0 and solve. This gave us a y value of 27.5, leading to the correct answer of 22.5

The thing that is troubling us is the following: why is it that the discriminant being =0 leads to the greatest value of y?

The discriminant tells us that there is only one solution, but we can't make the link as to why this leads to the greatest value of y.

An explanation would be much appreciated by both of us. Thanks in advance.

So, you'll have ended up with the discriminant D being a function of y. And I would expect (without having done any calculations) that D is a decreasing function of y.

So if y0 satisfies D(y0) = 0, then if y > y0, D(y) < 0.

But if D(y) is < 0, there are no solutions. So there are no solutions for any value of y > y0.
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Sinnoh
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Original post by jc768
Hi, my friend and I were solving question 7:

https://ibb.co/5K26yLv

We have succeeded in getting through the problem.

However one thing is bugging us, in 7iii) we are asked to use the discriminant to find the greatest distance that the golf ball can clear the tree.

To do this it felt instinctive to us to set the discriminant equal to 0 and solve. This gave us a y value of 27.5, leading to the correct answer of 22.5

The thing that is troubling us is the following: why is it that the discriminant being =0 leads to the greatest value of y?

The discriminant tells us that there is only one solution, but we can't make the link as to why this leads to the greatest value of y.

An explanation would be much appreciated by both of us. Thanks in advance.

Here's how I think of it. When the discriminant is 0 it means that the curve is just touching the y intercept line at one specific point. If you set the discriminant is > 0 then the curve must go a little bit past the intercept and then come back for there to be two roots, in which case the maximum point isn't related to the height of the intercept. Similar issue with taking the discriminant < 0.
Original post by jc768
Hi, my friend and I were solving question 7:

https://ibb.co/5K26yLv

We have succeeded in getting through the problem.

However one thing is bugging us, in 7iii) we are asked to use the discriminant to find the greatest distance that the golf ball can clear the tree.

To do this it felt instinctive to us to set the discriminant equal to 0 and solve. This gave us a y value of 27.5, leading to the correct answer of 22.5

The thing that is troubling us is the following: why is it that the discriminant being =0 leads to the greatest value of y?

The discriminant tells us that there is only one solution, but we can't make the link as to why this leads to the greatest value of y.

An explanation would be much appreciated by both of us. Thanks in advance.

If you take a general quadratic function y = ax^2 + bx + c are rearrange it into "completed square" form, it is quite easy to see that the max (or min) value of y is given by -(b^2 - 4ac)/4a, which is obviously related to the discriminant. It may be that that this is what you are expected to do in part (iii). But to be clear, you have to consider the function obtained in part (i) as y = f(tan(alpha)) with x replaced by the constant value 150, rather than treating it as y = f(x).
I must admit I don't understand your approach of setting the discriminant equal to zero.
Original post by old_engineer
If you take a general quadratic function y = ax^2 + bx + c are rearrange it into "completed square" form, it is quite easy to see that the max (or min) value of y is given by -(b^2 - 4ac)/4a, which is obviously related to the discriminant. It may be that that this is what you are expected to do in part (iii). But to be clear, you have to consider the function obtained in part (i) as y = f(tan(alpha)) with x replaced by the constant value 150, rather than treating it as y = f(x).
I must admit I don't understand your approach of setting the discriminant equal to zero.

What I'd expect:
First you form the quadratic function y(x) representing the height, y, as a function of horizontal distance, x.
To just clear a tree of height H at a distance d, we must have y(d) = H.
Substitute in these values and then rearrange as a new quadratic in tan⁡α\tan\alpha (where the coefficients of this quadratic will be functions of d and H).
The solutions to this quadratic are the angles you'd have to choose to just clear the tree.
Without actually solving it, you know you can clear the tree if and only if the quadratic has solutions. That is, iff the discriminant is >= 0.
The maximum height H you can clear is therefore the maximum value of H such that the discriminant is >= 0, which is going to end up being the value of H that makes the discriminant equal to 0.
(edited 3 years ago)

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