- #1

- 24

- 0

## Homework Statement

Derive an equation for the horizontal range of a projectile with a landing point at a different altitude from its launch point. Write the equation in terms of the initial velocity, the acceleration due to gravity, the launch angle, and the vertical component of the displacement. Please check to see if it is correct!

I tried to make the formulas as realistic as possible.

p.s. the "/" tend's to mean a fraction. a/b =

^{a}/

_{b}

## Homework Equations

x = vcosΘt, t = v/cosΘ (1)

y = vsinΘt - 1/2 gt

^{2}(2)

## The Attempt at a Solution

Solving (1) for t and substituting this expression in (2) gives:

y = xtanΘ - (gx

^{2}/2v

^{2}cos

^{2}Θ)

y = xtanΘ - (gx

^{2}sec

^{2}Θ/2v

^{2}) .....(Trig. identity)

y = xtanΘ - (gx

^{2}/2v

^{2}) * (1+tan

^{2}Θ) .....(Trig. identity)

0 = (-gx

^{2}/2v

^{2}* tan

^{2}Θ) + xtanΘ - (gx

^{2}/2v

^{2}) - y .....(Algebra)

Let p = tanΘ

0 = (-gx

^{2}/2v

^{2}* p

^{2}) + xp - (gx

^{2}/2v

^{2}) - y .....(Substitution)

p =

__-x ± √{ x__.....(Quadratic formula)

^{2}- 4(-gx^{2}/2v^{2})(-gx^{2}/2v^{2}-y)},,,,,,,,,,,,,,,,,,,2 (-gx

^{2}/2v

^{2})

p =

__v__.....(Algebra)

^{2}± √{ v^{4}- g(gx^{2}+ 2yv^{2})},,,,,,,,,,,,,,,,,,,,,,,,,gx

tanΘ =

__v__.....(Substitution)

^{2}± √{ v^{4}- g(gx^{2}+ 2yv^{2})},,,,,,,,,,,,,,,,,,,,,,,,,gx

Θ =

__tan__

^{-1}[v^{2}± √{ v^{4}- g(gx^{2}+ 2yv^{2})}],,,,,,,,,,,,,,,,,,,,,,,,,gx