Projectile Motion, different starting/ending height.

[TrueBlood]

If you are given the projectile angle, initial velocity, and height you can find the projectile range using the quadratic equation (solving for t). One thing I don't understand is why the quadratic equation seems to be the only way to solve these problems.

For instance, let's say an object is shot 8m/s, 40deg over the horizon, and at a height of 2.4 m. Now finding the range is straightforward.

(change in y) = Vot + 1/2at^2
-2.4 = Sin40*8*t + 1/2(-10)t^2
-2.4=5.12t + -5t^2
-5t^2+5.12t-2.4 = 0 This is the quadratic equation, which you solve for t


Now let's say you didn't want to use the quadratic equation.
(1) You know that the object travels 6.1 m in 1 sec.
Vf = Sin40*8 + (-10)t
0 = 5.12 -10t
tup = 0.5s
total projectile time (back to it's original starting height) = 1 s
.:. (change in x) = cos(40)*8*1s = 6.1 m
(2) Next let's find the velocity at the point, after 1 sec.
V(1 sec) = Sin(40)*8 + (-10)t
V(1 sec) = 5.12 -10
V(1 sec) = -4.88 This, of course, is the y-direction velocity only
(3) Now let's see what the final velocity is, at -2.4 m below it's starting point
V^2 = Vo^2 + 2a(change in y)
V^2 = (-4.88^2) + 2(-10)(-2.4) Vo is at the 1 second point, Vf is after that at a -2.4 height
V^2 = 23 - 48
V^2 = -25 Ok I get a negative number... no good, so I'll change it to a positive number even though that is mathematical sin
V = 5 m/s
(4) Now let's see how long it takes to get from -4.88 to -5m/s with gravity
Vf = Vo + at
-5 = -4.88 + (-10)t
t = 0.012
(5) Now add the calculated time (From 1 sec to end point) to the time it takes the projectile to get from starting point to 1 sec.
1 + 0.012 = TotalTime
(6) Now range is easy
(Change in x) = Cos40*8*1.012
(Change in x = 6.2 m

This of course is wrong The actual time it adds to fall from 1sec to final-place is 0.372 sec (or 1.372 sec total). You get that with the quadratic equation... making the actual range 8.3 m.

So why doesn't my ingenious method of creating more work for myself not work?

 

[TrueBlood]

Figured out a way to do it.

First find the max height , then the time it takes to fall from the max height to the ground (so height above original place + 2.4). Then add that time with the time it takes to reach the top. Then simply find range.