Trying to fix a simplification issue for a general range formula

[FallenLeibniz]

So I have derived a formula for the range of projectile in a vacuum fired from an arbitrary height h from above the ground as the following...

R=(vcosθ/g)[vsinθ+sqrt((vsinθ)^2+2gh)]Now it seems all well and good, but when I try to play with it by multiplying the RHS by the congugate term over the conjugate term (i.e. multiplying by one), I keep coming up with the 2gh factor on top which if h=0 leads to a discrepancy between the two. Can anyone help reconcile this? Note that all factors in the equation are positive values. I can provide a derivation of this equation later if neccessary, but am late to be somewhere and so I must do it later.

 

[haruspex]

It's not there's a discrepancy. If you plug in h = 0 you will 0/0, which is indeterminate. So the right thing to do is take the limit as h tends to 0 from above, and that will give you sin(2θ)/g, as before.

 

[FallenLeibniz]

Ok, I see the discontinuity now. I feel horrifically blind now.

 

[FallenLeibniz]

Thank you.

 

[PJC01]

I would suggest taking a step back and carefully reviewing your derivation to ensure that all the steps are correct. It is possible that there may be a simplification error or a missing term that is causing the discrepancy.

Additionally, I would recommend checking your assumptions and the conditions under which the formula was derived. Is the formula valid for all values of h, or are there certain restrictions? Are there any external factors that could affect the accuracy of the formula, such as air resistance or wind?

If the formula is indeed correct and valid, then it may be helpful to approach the problem from a different angle. Instead of trying to manipulate the formula algebraically, you could use numerical methods or simulations to test its accuracy for different values of h and compare the results.

In any case, it is important to thoroughly understand the derivation and assumptions of the formula in order to address any discrepancies and ensure its accuracy.