Introduction to Motion in Several Dimensions

In the subject "Motion in Physics" we studied what we mean in Physics by motion. We have defined the kinematic magnitudes (position, velocity and acceleration) that allow us to analyze and predict the behavior of a body in motion, whether it is an airplane, a ball or a satellite. Lastly, we have studied some simple motions in one dimension, as are: uniform rectilinear motion (u.r.m.) and uniformly accelerated rectilinear motion (u.a.r.m.), their equations and their graphs pointing out how to do the calculation of position, velocity and acceleration under these circumstances.

The truth is that, from these simple motions, it is possible to study other more complex motions that frequently occur in Nature. For example, when a football player hit the ball with his head trying to score, the motion of the ball does not follow a straight trajectory, but rather a parable that can be described as the composition of a uniform rectilinear motion and a uniformly accelerated rectilinear motion.

Superposition principle in motion

It was Galileo the first one to realize that a complex motion can be studied as a composition of other simpler ones: this is known as superposition principle which is used in other areas of Science. Applied to kinematics:

To study motion occurring in multiple dimensions as the superposition of other simpler ones, we take the following steps:

• We determine the type of each component motion that contribute to the more complex resulting motion. For example, in case of the soccer head-butt in the previous figure, it would be a uniform rectilinear motion in the horizontal axis and a uniformly accelerated rectilinear motion or uniformly varied rectilinear motion (free fall) in the vertical axis
• We solve each motion with the kinematic equations used for each component motion

• We apply the superposition principle. This way the kinematic magnitudes will be:

  ${\overrightarrow{r}}_T={\overrightarrow{r}}_1+{\overrightarrow{r}}_2$ ;  ${\overrightarrow{v}}_T={\overrightarrow{v}}_1+{\overrightarrow{v}}_2$  ; ${\overrightarrow{a}}_T={\overrightarrow{a}}_1+{\overrightarrow{a}}_2$

Motion in two and three dimensions

The previous expressions and examples correspond to motion in two dimensions because they are the most common ones: notice that they only have two components.

However, there are cases in which all three coordinates of the position vector change. These cases present a greater mathematical complexity, but the indicated procedures are equally valid in three dimensions. Thus, the kinematic magnitudes will have three components in these cases:,>

$${\overrightarrow{r}}_T={\overrightarrow{r}}_1+{\overrightarrow{r}}_2+{\overrightarrow{r}}_3 ; {\overrightarrow{v}}_T={\overrightarrow{v}}_1+{\overrightarrow{v}}_2+{\overrightarrow{v}}_3 ; {\overrightarrow{a}}_T={\overrightarrow{a}}_1+{\overrightarrow{a}}_2+{\overrightarrow{a}}_3$$

Are you ready?