Acceleration #5

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Today, we get an artillery on a cliff above the same camp as last time. Osama was spotted through a window seated on the toilet in his outhouse 400 meters away and 100 meters down from our spot. This time, our artillery is better than the one we used last time, firing shells at 100m/s. The question is, how high do we aim it? We’ll need a new formula for that.

0=as_x²/v²•tan²θ+s_xtanθ+ as_x²/v²-s_y

You can see how I derived that for yourself if you want. It uses the same variables as before, so now to plug the numbers in.

0=-9.8•400²/100²•tan²θ+400tanθ+(-9.8•400²/100²-(-100))
↓
0=-9.8•160000/10000•tan²θ+400tanθ+(-9.8•160000/10000+100)
↓
0=-9.8•16tan²θ+400tanθ+(-9.8•16+100)
↓
0=-156.8tan²θ+400tanθ-56.8

Unfortunately, I don’t see it being easily factorable, so we put it through the quadratic formula.

(-b±√(b²-4ac))/2a

a=-156.8
b=400
c=-56.8

Let’s start with the addition version of it.

tanθ=(-400+√(160000-(4•-156.8•-56.8)))/(2•-156.8)
↓
tanθ=(-400+√(160000-35624.96))/-313.6
↓
tanθ=(-400+√124375.04)/-313.6
↓
θ=8.582852416°

Now the subtraction version.

tanθ=(-400-√(160000-(4•-156.8•-56.8)))/(2•-156.8)
↓
tanθ=(-400-√(160000-35624.96))/-313.6
↓
tanθ=(-400-√124375.04)/-313.6
↓
θ=67.38090412°

Well, it would’ve been nice to have one of the answers be obviously wrong, but it doesn’t always work that way. Let’s check both answers.

θ=8.582852416°
↓
tanθ=(-400+√124375.04)/-313.6
↓
0=-156.8tan²θ+400tanθ-56.8

You put the value of tanθ into the equation on the third line of that set, use your calculator, and you get

0=0

…which means θ=8.582852416° is a right answer. Wait, we’re not quite done yet. It’s always a good idea to check the other answer, too.

θ=67.38090412°
↓
tanθ=(-400-√124375.04)/-313.6
↓
0=-156.8tan²θ+400tanθ-56.8

You do the same thing again, putting the answer you got for tanθ into the equation. This time you get

0=0.000000002

Close, but not good enough to be a right answer.

Now we can aim the artillery 8.582852416° above horizontal, fire, and watch Osama’s outhouse get blown up.