2010-12-22, 09:35 PM
Assuming parabolic motion, the horizontal velocity vector of a projectile is v(0) cos(θ
. The vertical velocity vector is modeled by v(0) sin(θ
t + gt. Therefore, we integrate to find the positions:
∫ v(t) dt = ∫ v(0) cos(θ
dt = x(t)
v(0) cos(θ
t +x(0) = x(t)
∫ v(t) dt = ∫ v(0) sin(θ
+gt dt = y(t)
v(0) sin(θ
t +gt^2/2 +r(0) = y(t)
Since x(0) is 0 at my starting point, I'm going to rewrite this as:
v(0) cos(θ
t = x(t)
I'm trying to find a general solution that allows for me to clear a wall of height h distance d away. Thus, x(t) = d and y(t) = h. Therefore, my problem is to eliminate t. If I rearrange x(t), I get:
t = d /(v(0) cos(θ
)
Subbing into y(t), I get:
g(d /(v(0) cos(θ
))^2/2 +v(0) sin(θ
(d /(v(0) cos(θ
)) +r(0) = h
g(d /(v(0) cos(θ
))^2/2 +v(0) sin(θ
(d /(v(0) cos(θ
)) +r(0) -h = 0
Because I have to clear the wall:
g (d /(v(0) cos(θ
) )^2/2 +v(0) sin(θ
(d /(v(0) cos(θ
) ) +r(0) -h > 0
Simplifying:
gd^2 (sec(θ
)^2 /(2 v(0)^2) + d tan(θ
+ r(0) -h > 0
gd^2 (1+(tan(θ
)^2) /(2 v(0)^2) + d tan(θ
+ r(0) -h > 0
gd^2 (tan(θ
)^2 /(2 v(0)^2) + d tan(θ
+ r(0) -h + gd^2 /(2v(0)^2) > 0
Now I can use the quadratic formula to solve for tan(θ
.
tan(θ
> ( -d +/- √(d^2 -4(gd^2/(2v(0)^2)(gd^2 /(2v(0)^2) +r(0) -h)) ) /(gd^2 /v(0)^2)
tan(θ
> ( -d +/- √(d^2(1 -(2g/v(0)^2)(gd^2 /(2v(0)^2) +r(0) -h)) ) /(gd^2 /v(0)^2)
tan(θ
> ( -d +/- d√((1 -(2g/v(0)^2)(gd^2 /(2v(0)^2) +r(0) -h)) ) /(gd^2 /v(0)^2)
tan(θ
> (v(0)^2) ( -1 +/- √((1 -(2g/v(0)^2)(gd^2 /(2v(0)^2) +r(0) -h)) ) /(gd)
tan(θ
> (-v(0)^2 +/- v(0)√((v(0)^2 -(2g)(gd^2 /(2v(0)^2) +r(0) -h)) ) /(gd)
tan(θ
> (-v(0)^2 +/- v(0)√((v(0)^2 -(gd/v(0))^2 -(2g)(r(0) -h)) ) /(gd)
θ > arctan((-v(0)^2 +/- v(0)√((v(0)^2 -(gd/v(0))^2 -(2g)(r(0) -h)) ) /(gd)
When I put this into my calculator, however, it is giving me incorrect solutions. I don't know what I did wrong.
. The vertical velocity vector is modeled by v(0) sin(θ
t + gt. Therefore, we integrate to find the positions:∫ v(t) dt = ∫ v(0) cos(θ
dt = x(t)v(0) cos(θ
t +x(0) = x(t)∫ v(t) dt = ∫ v(0) sin(θ
+gt dt = y(t)v(0) sin(θ
t +gt^2/2 +r(0) = y(t)Since x(0) is 0 at my starting point, I'm going to rewrite this as:
v(0) cos(θ
t = x(t)I'm trying to find a general solution that allows for me to clear a wall of height h distance d away. Thus, x(t) = d and y(t) = h. Therefore, my problem is to eliminate t. If I rearrange x(t), I get:
t = d /(v(0) cos(θ
)Subbing into y(t), I get:
g(d /(v(0) cos(θ
))^2/2 +v(0) sin(θ
(d /(v(0) cos(θ
)) +r(0) = hg(d /(v(0) cos(θ
))^2/2 +v(0) sin(θ
(d /(v(0) cos(θ
)) +r(0) -h = 0Because I have to clear the wall:
g (d /(v(0) cos(θ
) )^2/2 +v(0) sin(θ
(d /(v(0) cos(θ
) ) +r(0) -h > 0Simplifying:
gd^2 (sec(θ
)^2 /(2 v(0)^2) + d tan(θ
+ r(0) -h > 0gd^2 (1+(tan(θ
)^2) /(2 v(0)^2) + d tan(θ
+ r(0) -h > 0gd^2 (tan(θ
)^2 /(2 v(0)^2) + d tan(θ
+ r(0) -h + gd^2 /(2v(0)^2) > 0Now I can use the quadratic formula to solve for tan(θ
.tan(θ
> ( -d +/- √(d^2 -4(gd^2/(2v(0)^2)(gd^2 /(2v(0)^2) +r(0) -h)) ) /(gd^2 /v(0)^2)tan(θ
> ( -d +/- √(d^2(1 -(2g/v(0)^2)(gd^2 /(2v(0)^2) +r(0) -h)) ) /(gd^2 /v(0)^2)tan(θ
> ( -d +/- d√((1 -(2g/v(0)^2)(gd^2 /(2v(0)^2) +r(0) -h)) ) /(gd^2 /v(0)^2)tan(θ
> (v(0)^2) ( -1 +/- √((1 -(2g/v(0)^2)(gd^2 /(2v(0)^2) +r(0) -h)) ) /(gd)tan(θ
> (-v(0)^2 +/- v(0)√((v(0)^2 -(2g)(gd^2 /(2v(0)^2) +r(0) -h)) ) /(gd)tan(θ
> (-v(0)^2 +/- v(0)√((v(0)^2 -(gd/v(0))^2 -(2g)(r(0) -h)) ) /(gd)θ > arctan((-v(0)^2 +/- v(0)√((v(0)^2 -(gd/v(0))^2 -(2g)(r(0) -h)) ) /(gd)

When I put this into my calculator, however, it is giving me incorrect solutions. I don't know what I did wrong.
