Two-Dimensional Kinematics: Motion in a Plane

This article outlines the fundamental concepts necessary to analyze the motion of objects in two dimensions, without regard to the forces that cause the acceleration involved. It assumes a familiarity with one-dimensional kinematics, as it expands the same concepts into a two-dimensional vector space.

First Step: Choosing Coordinates

Kinematics involves displacement, velocity, and acceleration which are all vector quantities that require both a magnitude and direction. Therefore, to begin a problem in two-dimensional kinematics you must first define the coordinate system you are using. Generally it will be in terms of an x-axis and a y-axis, oriented so that the motion is in the positive direction, although there may be some circumstances where this is not the best method.

In cases where gravity is being considered, it is customary to make the direction of gravity in the negative-y direction. This is a convention that generally simplifies the problem, although it would be possible to perform the calculations with a different orientation if you really desired.

Velocity Vector

The position vector r is a vector that goes from the origin of the coordinate system to a given point in the system. The change in position (delta-r) is the difference between the start point (r₁) to end point (r₂). We define the average velocity (v_av) as:

v_av = (r₂ - r₁) / (t₂ - t₁) = delta-r/delta-t

Taking the limit as delta-t approaches 0, we achieve the instantaneous velocity v. In calculus terms, this is the derivative of r with respect to t, or dr/dt.

As the difference in time reduces, the start and end points move closer together. Since the direction of r is the same direction as v, it becomes clear that the instantaneous velocity vector at every point along the path is tangent to the path.

Velocity Components

The useful trait of vector quantities is that they can be broken up into their component vectors. The derivative of a vector is the sum of its component derivatives, therefore:

v_x = dx/dt
v_y = dy/dt

The magnitude of the velocity vector is given by the Pythagorean Theorem in the form:

|v| = v = sqrt (v_x² + v_y²)

The direction of v is oriented alpha degrees counter-clockwise from the x-component, and can be calculated from the following equation:

tan alpha = v_y / v_x

Acceleration Vector

Acceleration is the change of velocity over a given period of time. Similar to the analysis above, we find that it's delta-v/delta-t. The limit of this as delta-t approaches 0 yields the derivative of v with respect to t.

In terms of components, the acceleration vector can be written as:

a_x = dv_x/dt
a_y = dv_y/dt

or

a_x = d²x/dt²
a_y = d²y/dt²

The magnitude and angle (denoted as beta to distinguish from alpha) of the net acceleration vector are calculated with components in a fashion similar to those for velocity.

Working with Components

Frequently, two-dimensional kinematics involves breaking the relevant vectors into their x- and y-components, then analyzing each of the components as if they were one-dimensional cases. Once this analysis is complete, the components of velocity and/or acceleration are then combined back together to obtain the resulting two-dimensional velocity and/or acceleration vectors.

Three-Dimensional Kinematics

The above equations can all be expanded for motion in three dimensions by adding a z-component to the analysis. This is generally fairly intuitive, although some care must be made in making sure that this is done in the proper format, especially in regards to calculating the vector's angle of orientation.