Motion in 2D and 3D

Previous: Vectors

Motion in 2D and 3D is the third lecture in the Mechanics section of PH1011. It covers a 3D overview of previous 2D vectors, and projectile motion.

Next: Newton's Laws of Motion

Position and Velocity
Position vectors define the position of an object at any point in its motion through 3D space. This vector simply extends between the origin and the object.

The vector describing average velocity is a linear superposition of three 2D vectors, and is given as Δx/Δti + Δy/Δtj + Δz/Δtk.

The instantaneous velocity vector is limited as Δt -> 0, however is again a simple superposition of 2D motion concepts. v(t) = dr(t)/dt = vx(t)i + vy(t)j + vz(t)k.

Speed, Distance and Acceleration
The instantaneous speed is given by the magnitude of the velocity. In small time intervals the distance travelled is v(t)dt - therefore a total distance is the integral of v between the start and end times (d = ∫titfv(t)dt).

The average acceleration vector is similar to the average velocity vector - Δx/Δti + Δvy/Δtj + Δvz/Δtk. The instantaneous acceleration vector is the sum of linear vectors again - a = ax(t)i + ay(t)j + az(t)k.

Projectile Motion
Projectile motion is usually considered in only two dimensions, and can be described by a vertical superposition on the y axis and a horizontal motion on the x axis. The origin point of a projectile is usually taken to be where it has been fired from; from there it follows a trajectory dependent on the effects of gravity and air resistance. However, assumptions can be made to produce a simplified model:
 * The object is a point mass
 * Air resistance and planetary curvature/rotation is negligible
 * Gravitational force is constant over short distances

Neglecting air resistance causes gravity to be the only force acting on the mass. This gives an acceleration vector of -gj. The three kinematic equations can be used to derive further equations for the mass's time of flight, trajectory shape, range and maximum height.

Summary
Position vectors are the vecor between an object's location and its origin. Most 3D vectors are merely superpositions of their three linear components. Projectile motion uses simplifications to allow easy calculation of the path of a projectile.