2.4 Accelerated motion in two dimensions

We have already studied two dimensional motions such as projectile and uniform circular motion. These motions are the most celebrated examples of two dimensional motion, but it is easy to realize that they are specific instances with simplifying assumptions. The motions that we investigate in our surrounding mostly occur in two or three dimensions in a non-specific manner. The stage is, therefore, set to study two-dimensional motion in non-specific manner i.e. in a very general manner. This requires clear understanding of both linear and non-linear motion. As we have already studied circular motion - an instance of non-linear motion, we can develop an analysis model for a general case involving non-linear motion.

The study of two dimensional motion without any simplifying assumptions, provides us with an insight into the actual relationship among the various motional attributes, which is generally concealed in the consideration of specific two dimensional motions like projectile or uniform circular motion. We need to develop an analysis frame work, which is not limited by any consideration. In two dimensional motion, the first and foremost consideration is that acceleration denotes a change in velocity that reflects a change in the velocity due to any of the following combinations :

• change in the magnitude of velocity i.e. speed
• change in the direction of velocity
• change in both magnitude and direction of velocity

In one dimensional motion, we mostly deal with change in magnitude and change in direction limited to reversal of motion. Such limitations do not exist in two or three dimensional motion. A vector like velocity can change by virtue of even direction only as in the case of uniform circular motion. Further, a circular motion may also involve variable speed i.e. a motion in which velocity changes in both direction and magnitude.

Most importantly, the generalized consideration here will resolve the subtle differences that arises in interpreting vector quantities like displacement, velocity etc. We have noted that there are certain subtle differences in interpreting terms such as Δr and |Δ r |; dr/dt and |d r /dt|; dv/dt and |d v /dt| etc. In words, we have seen that time rate of change in the magnitude of velocity (speed) is not equal to the magnitude of time rate of change in velocity. This is a subtle, but significant difference that we should account for. In this module, we shall find that time rate of change in the magnitude of velocity (speed), as a matter of fact, represents the magnitude of a component of acceleration known as "tangential acceleration".

Characteristics of two dimensional motion

Let us have a look at two dimensional motions that we have so far studied. We observe that projectile motion is characterized by a constant acceleration, “g”, i.e. acceleration due to gravity. What it means that though the motion itself is two dimensional, but acceleration is one dimensional. Therefore, this motion presents the most simplified two dimensional motion after rectilinear motion, which can be studied with the help of consideration of motion in two component directions.