Question

show that flight time can be estimated by t=sqrt ((2h-2Rtan a)/g)

Answer 1

What's the question that leads to this?

Answer 2

There isn't one but the information that i have is that the velocity vector is <vcos a, vtsin a -gt> the position vector is <vtcos a, vtsin a -(1/2)gt^2 +h>.

R=(v^2 sin 2a)/g

t=(2vsin a)g

h=(v^2 sin^2 a)/g

Answer 3

Oh, okay, this looks like projectile motion. Just give me a sec.

Answer 4

I'll just tell you what I'm doing and you can do it too - you're being asked to find the time of flight. Given your equations, this is going to be the time elapsed between being fired (t=0, y=h) and when it strikes the ground (t=?, y=0).

Answer 5

k

Answer 6

Are you sure there's a minus between the 2h and 2Rtan(a)?

Answer 7

yep

Answer 8

The problem is, if there is a minus instead of a plus, that expression you have is complex. When you evaluate the radicand, it's negative, so you end up with an imaginary number.

Answer 9

\[h-R \tan \alpha=\frac{v^2\sin^2 \alpha}{g}-\frac{v^2\sin 2 \alpha}{g}\tan \alpha\]\[=\frac{v^2\sin^2 \alpha -v^2(2\sin \alpha \cos \alpha) }{g}\frac{\sin \alpha}{\cos \alpha}\]\[\frac{-v^2\sin^2 \alpha}{g}\]Multiplying that by 2 and taking the square root will give you\[\sqrt{\frac{2h-2R \tan \alpha }{g}}=\sqrt{\frac{-2v^2 \sin ^2 \alpha}{g^2}}=\frac{\sqrt{2}v \sin \alpha}{g}i\]