Finding relationship between Range and Launch height

[Sam 1998]

Homework Statement

I am currently trying to find a way to determine the relationship between launch height and range for a projectile launched at less than horizontal.

Would vary launch height to and measure range.

I need a directly proportional equation or at least a linear relationship.

Homework Equations

In next part

The Attempt at a Solution

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Sh = Vh*t
Sv = Vov*t + 1/2*a*t^2

Where the launch velocity components are:

Vv = v*sin(launch angle)
Vh = v*cos(Launch Angle)

Therefore,

Sh = v * cos(launch angle) * t so t = Sh/v*cos(launch angle)
Sv = v*sin(launch angle) * t + 1/2*a*t^2

Substituting time,
Sv = v*sin(launch angle) * (Sh/ v * cos(launch angle)) + 1/2 * a * (Sh/ v * cos(launch angle))^2

This is as far as I have got however I need to find a way to show the direct relationship between launch height and range. I am assuming now this isn't possible due to the quadratic, however can anyone think of a solution?

 

[olivermsun]

Can you solve for the zeros of a quadratic equation (in general)?

 

[Sam 1998]

Can you solve for the zeros of a quadratic equation (in general)?

Yes but having it in a quadratic form won't give a linear relationship. Is there an alternative to my method that would yield a linear relationship between range and launch height?

 

[phinds]

Yes but having it in a quadratic form won't give a linear relationship. Is there an alternative to my method that would yield a linear relationship between range and launch height?

Why would you expect to have the relationship be linear? I'm not saying that I KNOW it to be non-linear, but I would have started out with the assumption that it would NOT be, not that it would be.

 

[Sam 1998]

Why would you expect to have the relationship be linear? I'm not saying that I KNOW it to be non-linear, but I would have started out with the assumption that it would NOT be, not that it would be.

For the purpose of the school report, we must have a linear relationship in the form y = mx +c, so that the m value can be compared to measured values to prove the accuracy of the data.

 

[phinds]

For the purpose of the school report, we must have a linear relationship in the form y = mx +c, so that the m value can be compared to measured values to prove the accuracy of the data.

Well, maybe the relationship IS linear, I just would not have expected it to be.

 

[olivermsun]

For the purpose of the school report, we must have a linear relationship in the form y = mx +c, so that the m value can be compared to measured values to prove the accuracy of the data.

Why must the relationship be linear to compare measured to predicted values? You could have some other functional form and still compare, couldn't you?