5.1: Introduction to the Vector Description of Motion in Two Dimensions

Where was the chap I saw in the picture somewhere? Ah yes, in the dead sea floating on his back, reading a book with a parasol open. Couldn’t sink if you tried: so thick with salt. Because the weight of the water, no, the weight of the body in the water is equal to the weight of the what? Or is it the volume equal to the weight? It’s a law something like that. Vance in High school cracking his fingerjoints, teaching. The college curriculum. Cracking curriculum. What is weight really when you say weight? Thirtytwo feet per second per second. Law of falling bodies: per second per second. They all fall to the ground. The earth. It’s the force of gravity of the earth is the weight.

~ James Joyce

We have introduced the concepts of position, velocity and acceleration to describe motion in one dimension; however we live in a multidimensional universe. In order to explore and describe motion in more than one dimension, we shall study the motion of a projectile in two-dimension moving under the action of uniform gravitation.

We extend our definitions of position, velocity, and acceleration for an object that moves in two dimensions (in a plane) by treating each direction independently, which we can do with vector quantities by resolving each of these quantities into components. For example, our definition of velocity as the derivative of position holds for each component separately. In Cartesian coordinates, the position vector \(\overrightarrow{\mathbf{r}}(t)\) with respect to some choice of origin for the object at time t is given by

\[\overrightarrow{\mathbf{r}}(t)=x(t) \hat{\mathbf{i}}+y(t) \hat{\mathbf{j}} \nonumber \]

The velocity vector \(\overrightarrow{\mathbf{v}}(t)\) at time t is the derivative of the position vector,

\[\overrightarrow{\mathbf{v}}(t)=\frac{d x(t)}{d t} \hat{\mathbf{i}}+\frac{d y(t)}{d t} \hat{\mathbf{j}} \equiv v_{x}(t) \hat{\mathbf{i}}+v_{y}(t) \hat{\mathbf{j}} \nonumber \]

where \(v_{x}(t) \equiv d x(t) / d t\) and \(v_{y}(t) \equiv d y(t) / d t\) denote the x - and y -components of the velocity respectively.

The acceleration vector \(\overrightarrow{\mathbf{a}}(t)\) is defined in a similar fashion as the derivative of the velocity vector,

\[\overrightarrow{\mathbf{a}}(t)=\frac{d v_{x}(t)}{d t} \hat{\mathbf{i}}+\frac{d v_{y}(t)}{d t} \hat{\mathbf{j}} \equiv a_{x}(t) \hat{\mathbf{i}}+a_{y}(t) \hat{\mathbf{j}} \nonumber \]

where \(a_{x}(t) \equiv d v_{x}(t) / d t\) and \(a_{y}(t) \equiv d v_{y}(t) / d t\) denote the x- and y-components of the acceleration