# Formula for vertical and horizontal components of trajectory

Oleg Volk

August 9, 2006, 05:10 PM

I need to come up with a formula for x and y coordinates of a ballistic projectile, neglecting air resistance. I forgot most of my physics and math, so I'd like to verify my guesses:

for vertical position (sinLAUNCH ANGLE * velocity - FREEFALL ACCELERATION * time) * time

for horizontal cosineLAUNCH ANGLE * velocity * time

Thanks in advance...

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Gonefission

August 9, 2006, 05:15 PM

Check this out

http://en.wikipedia.org/wiki/Trajectory_of_a_projectile

The sad thing is I got through Calc 3 with some decent grades and I don't remember a thing. :(

Oleg Volk

August 9, 2006, 05:24 PM

I looked through that page earlier. Let's look here:

http://upload.wikimedia.org/math/d/9/3/d937be7f39b6e438b5e9a37da16db59b.png

v = velocity?

g = freefall acceleration?

And how do I find x? Is my original formula for that correct?

Update:

My father just emailed me the correct formula:

Almost right...

for vertical position (sinLAUNCH ANGLE * velocity - FREEFALL ACCELERATION * time / 2) * time reflecting that (sinLAUNCH ANGLE * velocity - FREEFALL ACCELERATION * time) is the vertical speed at that time but the average one was half of that.

Gonefission

August 9, 2006, 05:55 PM

Here is the legend from the top of that page:

* g: the acceleration due to gravity—usually taken to be 9.81 m/s2 near the Earth's surface

* θ: the angle at which the projectile is launched

* v: the velocity at which the projectile is launched

* y0: the initial height of the projectile

* d: the total horizontal distance travelled by the projectile

According to my TI-89 solving for x gets you the attached formula

WayneConrad

August 9, 2006, 06:17 PM

Oleg, if you're shooting on the moon, remember that you'll need a different value for g.

This could be the coolest range report ever.

johnmcl

August 10, 2006, 10:27 AM

Hi Oleg,

That last equation doesn't take into consideration a couple of trig properties that will simplify your work.

Essentially Range = V^2 sin(2*angle)/g

There's a great astrophysics page here that has done the mathematical heavy lifting:

http://hyperphysics.phy-astr.gsu.edu/hbase/traj.html

Now a couple of points. Express your angle in radians, and the effect of gravity (g) as 32 ft/sec^2. That will allow you to keep your muzzle velocity in feet per second.

Hope that helps?

John

roo_ster

August 10, 2006, 12:21 PM

Here's how I always did my work in English measurements:

1. Convert given data into mks

2. Solve equation via algebra/trig/calc

3. Plug in mks values & solve

4. Convert to English measurements

I love English measurements for everday life, but I want metric for engineering/physics/tech work.

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