# Finding relationship between Range and Launch height

1. Feb 8, 2017

### Sam 1998

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

In next part

3. The attempt at a solution

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?

2. Feb 8, 2017

### olivermsun

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

3. Feb 8, 2017

### Sam 1998

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?

4. Feb 8, 2017

### phinds

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.

5. Feb 8, 2017

### Sam 1998

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.

6. Feb 8, 2017

### phinds

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

7. Feb 9, 2017

### olivermsun

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?