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Oomph!
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Hello everyone. I have this problem:
1. The problem statement, all variables and given/known d
I throw stone from height h at an angle α, then at an angle β (so I know h, α, β).
I throw it with same velocity v. The stones fall in same distance d.
The question is: find the d (distance) and v (initial velocity).
So, this is projection at an angle. Coordinates of velocity and coordinates of position:
vx=v0cosα
vy=v0sinα-gt
x=tv0cosα
y=h+tv0sinα-0,5gt2
We can make the formula for the calculation of d:
d=(v2sin2α+sqrt(v4*(sin2α)2+8hgv2*(cosα)2)/2g
I could solve this equation for two angles (distance is same) and find v:
(v2sin2α+sqrt(v4*(sin2α)2+8hgv2*(cosα)2) =
(v2sin2β+sqrt(v4*(sin2β)2+8hgv2*(cosβ)2)
I have done it. However, the solution is not nice, really big formula. So, I think this is not good process. There must be something easier. I know the answer:
d=h*tan(α+β)
v2=0,5gh*sin(α+β)*tan(α+β)/(cosα*cosβ)
My solution was really difficult and there was not any way how to make it easier.
So, could you show me how to make better solution?
Thank you.
1. The problem statement, all variables and given/known d
I throw stone from height h at an angle α, then at an angle β (so I know h, α, β).
I throw it with same velocity v. The stones fall in same distance d.
The question is: find the d (distance) and v (initial velocity).
Homework Equations
So, this is projection at an angle. Coordinates of velocity and coordinates of position:
vx=v0cosα
vy=v0sinα-gt
x=tv0cosα
y=h+tv0sinα-0,5gt2
The Attempt at a Solution
We can make the formula for the calculation of d:
d=(v2sin2α+sqrt(v4*(sin2α)2+8hgv2*(cosα)2)/2g
I could solve this equation for two angles (distance is same) and find v:
(v2sin2α+sqrt(v4*(sin2α)2+8hgv2*(cosα)2) =
(v2sin2β+sqrt(v4*(sin2β)2+8hgv2*(cosβ)2)
I have done it. However, the solution is not nice, really big formula. So, I think this is not good process. There must be something easier. I know the answer:
d=h*tan(α+β)
v2=0,5gh*sin(α+β)*tan(α+β)/(cosα*cosβ)
My solution was really difficult and there was not any way how to make it easier.
So, could you show me how to make better solution?
Thank you.
Last edited: