# Basic plane kinematics

I never did physics in my life and this is really hard for me. I am currently trying to implement movement in the X and Y axis of a plane, in a video game I am making.

What I want to do is, given a starting angle and a velocity, and the value of gravity, calculate in different points of time the current angle and the displacement in X and Y.

So, lets say, my current velocity is 4 units / second and my current angle is 30 degrees and my gravity is 9.8 u/s, I want to pass an argument in seconds and calculate the final angle, velocity, and the amount of X and Y the body moved.

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• Gravity's acceleration is 9.8 u/s^2, sorry couldnt resist ;) Using a physics engine like bullet is not an option? – Maik Semder Jan 9 '13 at 8:55
• That would be useless, as I don't want to implement collisions and complicated movement. I am actually making a fan-made game of Super Mario 64 – Pacha Jan 9 '13 at 20:02

If the acceleration is constant, e.g. the only force acting on the body is gravitational, we can use a trivial expression for the displacement at any time. It is a integrated twice with respect to t. In vector form: x(t) = x(0) + v(0) t + ½ a.

The starting velocity can be derived from θ using: v = v (cos θ, sin θ) or a variant, depending on how the angle is defined.

With the example values of:

• x(0) = (0, 0)
• v(0) = 4
• θ = 30°
• a = (0, -9.8)

the displacement can be calculated as: x(t) = (2√3, 2) t + (0,-4.9) t².

Just do your math step by step (as long as you're not looking for complex simulation of this):

To move the plane based on its acceleration (plane.a; can be constant or 0) you should first calculate its velocity along the axes:

plane.vx += plane.a * cos(plane.angle) * delta_time / 2.0;
plane.vy += plane.a * sin(plane.angle) * delta_time / 2.0;


This should get you some basic movement. Based on your coordinate system (e.g. in which direction the y coordinate becomes bigger) you might have to negate one or more parameters.

Once this is done, just apply the gravity as well:

plane.vy += gravity * delta_time / 2.0;


Then you can just update your coordinates:

plane.x += plane.vx * delta_time;
plane.y += plane.vy * delta_time;


If you're using fixed time steps in your logic, you can just assume delta_time being 1, which makes the whole calculation even easier to be done. The velocities become simple deltas and your acceleration essentially becomes a constant velocity:

plane.x += plane.v * cos(plane.angle);
plane.y += plane.v * sin(plane.angle) + gravity;