# Show that the horizontal range is 4h/tan(theta)

1. Jan 27, 2014

### negation

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

A projectile launched at angle Θ to the horizontal reaches maximum height h. Show that its horizontal range is 4h/ tan Θ.

3. The attempt at a solution

tapex = $vi sin Θ/g$
tfull trajectory = $2vi sin Θ/g$
h(tapex) = h = $vi^2 sin^2 Θ/2g$

x(tfull trajectory) = x = $\frac{vi^2 sin (2Θ)}{g}$

2. Jan 27, 2014

### voko

Consider the ratio of range to max height.

3. Jan 27, 2014

### negation

I did. The expression for the full range and heigh, h is in the op.
But I'm unable to reduce them to the appropriate wxpression

4. Jan 27, 2014

### voko

Show what you get and how you get that.

5. Jan 27, 2014

### negation

I worked out hapex = vi ^2 sin^Θ and xfull range = vi^2(2Θ)/g
The above was obtained by substituting t/2 intp the y-displacement and t into the x-displacement.

6. Jan 27, 2014

### voko

As I said. Consider the ratio of range to max height.

7. Jan 27, 2014

### negation

I did.

The horizontal range is 2vi^2 sinΘcosΘ.
I simplified 4h/tanΘ to 2vi^2sinΘcosΘ/g.
I suppose this is a sufficient condition for the proof?

8. Jan 28, 2014

### voko

I cannot see your analysis of the ratio of range to max height. Which is strange, because you have found a formula for range, and a formula for max height. All you need is to divide one by another.

9. Jan 28, 2014

### negation

I failed to obtain the required equation. Why do we need the ratio? The question has not asked for that.

My method:
I found x = vi^2 sin(2Θ)/g = 2vi^2 sinΘcosΘ
I reduced 4h/tanΘ to = 2vi^2 sinΘcosΘ, where h = vi^2 sin^2Θ/2g and tan Θ = sinΘ/cosΘ.

10. Jan 28, 2014

### voko

You are required to prove that $$x = {4h \over \tan \theta}$$ That means $${x \over h} = {4 \over \tan \theta}$$

Regarding your "failure", how are $\cot$ and $\tan$ related?

11. Jan 28, 2014

### negation

cot Θ= 1/tanΘ
It never occurred to me I had to perform a x/h ratio. But anyway, answer found.

Last edited: Jan 28, 2014