Hi,
I am trying to map a trajectory path between two point. All I know is the two points in question and the distance between them. What I would like to be able to calculate is the velocity and angle necessary to hit the end point.
I would also like to be able to factor in some gravity and wind so that the path/trajectory is a little less 'perfect.' Its for a computer game.
Thanks F.
The angle is easy, atan2(pB.x-pA.x,pB.y-pA.y). The velocity vector should be (pB-pA)*speed. And to add gravity/wind (gravity is just wind with a negative y component) add the (scaled) wind vector to your velocity at every simulation tick (you're basically adding it as acceleration).
Just a link, sorry : http://www.gamedev.net/reference/articles/article694.asp.
There is a lot of papers about game physics at gamedev. Have a look.
(BTW : wind only add some velocity to an object. The hard part is gravity.)
The book:
Modern Exterior Ballistics: The Launch and Flight Dynamics of Symmetric Projectiles ISBN-13: 978-0764307201
Is the modern authority on ballistics. For accuracy you'll need the corrections:
http://www.dexadine.com/mccoy.html
If you need something free and less authoritative, Dr. Mann's 1909 classic The bullet's flight from powder to target is available on books.google.com.
-kmarsh
PS Poor ballistics in games is a particular pet peeve of mine, especially the "shoots flat to infinity" ballistic model.
This entire physical situation can be described using the SUVAT equations of motion, since the acceleration at all times is constant.
The following explanation presumes understanding of basic algebra and vector maths. If you're not familiar with it, I strongly recommend you go read up on it before attempting to write the sort of game you have proposed. It also assumes you're dealing with 2D, though if you're dealing with 3D most of the same applies, since it's all in vector form - you just end up solving a cubic instead of quadratic, for which it may be best to use a numerical solver.
(Note: vectors represented in bold.)
Basically, you'll want to start by formulating your equation for displacement (in vector form):
r = ut + (at^2)/2
r is the displacement relative to the start position, u is the initial velocity, a is the acceleration (constant at all times). t is of course time.
a is dependent on the forces present in your system. In the general case of gravity and wind:
a = F_w/m - g j
where i is the unit vector in the x direction and j the unit vector in the y direction. g is the acceleration due to gravity (9.81 ms^-2 on Earth). F_w is the force vector due to the wind (this term disappears for no wind) - we're assuming this is constant for the sake of simplicity. m is the mass of the projectile.
Then you can simply substitute the equation for a into the equation for r, and you're left with an equation of three variables (r, u, t). Next, expand your single vector equation for r into two scalar equations (for x and y displacement), and use substitution to eliminate t (maths might get a bit tricky here). You should be left with a single quadratic equation with only r and u as free variables.
Now, you want to solve the equation for r = [target position] - [start position]. If you pick a certain magnitude for the initial velocity u (i.e. speed), then you can write the x and y components of u as U cos(a) and U sin(a) respectively, where U is the initial speed, and a the initial angle. This can be rearranged and with a bit of trignometry, you can finally solve for the angle a, giving you the launch velocity!
Most of the above description should be worked out on paper first. Then, it's simply a matter of writing a function to solve the quadratic formula and apply some inverse trigonometric functions to get the result.
P.S. Sorry for all the maths/physics in this post, but it was unavoidable! The OP seemed to be asking more about the physical rather than computational side of this, anyway, so that's what I've provided. Hopefully this is still useful to both the OP and other people.
As people have mentioned, while figuring out the angles between the points is relatively easy, determining the way that wind and gravity will affect the shot is more difficult.
Wind and gravity are both accelrating forces, though they act somewhat differently.
Gravity is easier, since it has both a constant direction (down) and magnitude regardless of the object. (Assuming that you're not shooting things with ridiculously high velocities). To calculate how gravity will affect the velocity of your object, just take the time since you last updated the velocity of the object, multiply it by your gravitational factor, and add it to your current velocity vector.
As a simple example, let's think of an object that is moving with a velocity of (3, 4, 7) in the x, y, z directions, with z being parallel with the force of gravity. You decide that your gravity value is -.3 You are ready to calculate the new velocity. When you check, you discover that 10 time units have passed since your last calculation (whatever your time units are...perhaps ticks or something). You take your time units (10), multiply by your gravity (-.3), which gives you -3. You add that to your Z, and your new velocity is (3, 4, 4). That's it. (This has been very simplified, but that should get you started.)
Wind is a bit different, if you want to do it right. If you want to do it a simple and easy way, you can make it like gravity...a constant force in a particular direction. But a more realistic way is to have the force be dependent on your current velocity vector. Put simply: if you're moving exactly with the wind, it shouldn't impart any force onto you. In this case, you simply calculate the magnitude of the force as the difference between its direction and your own.
A simple example of this might be if you were moving at (3, 0, 0), and the wind was moving at (5, 0, 0), and we can give the wind a strength of .5. (You also have to multiply by the time elapsed...for the sake of this example, to keep it simple, we'll leave the time-elapsed factor at 1) You calculate the difference in the vectors and multiply by your time difference (1), and discover that the difference is (2, 0, 0). You then multiply that vector by the wind strength, .5, and you discover that your velocity change is (1, 0, 0). Add that to your previous velocity, and you get (4, 0, 0)...so the wind has sped the object up slightly. If you waited another single time unit, you would have a difference of (1, 0, 0), multiplied by your strength of .5, so your final velocity would then be (4.5, 0, 0). As you can see, the wind provides less force as you become closer to it in velocity.) This is kind of neat, but may be overly complex for game ballistics.