In this lecture we will answer some of these (rather philosophical) questions. Motion involves change with time. In particular, we say an object undergoes motion if its position in space changes with time. Already we find trouble with this "definition": a spinning disk is clearly moving but its position does not change. So we must distinguish between translational motion and rotational motion. In this lecture we will study translational motion along one dimension. To study rotational motion one needs at least two spatial dimensions (in order to have a plane of rotation).
So change in position with time. How do we specify the position of an object? First we need two things: a coordinate system and a reference point. The coordinate system comes with units to measure distance. The position is then defined as the distance from a reference point. For example, we can choose the coordinate system to be the line of real numbers and the reference point to be the origin (the point $x = 0$). Given a real number we can mark it on the real line. Some of these numbers are to the left of the origin and some are to the right. This notion is called the orientation. If we just say that an object is 10 units from the origin then the position is ambiguous: 10 units to the left or to the right?
When an object moves, its position changes with time. The rate of change of position is called the velocity. The velocity of an object tells us how the position is changing. Change can be small or large, and positive or negative. For example, a small negative velocity means the object is moving slowly to the left, while a large positive velocity means the object is moving fast to the right. The magnitude of the velocity is called the speed.
Here is an example: the train between Stony Brook and Port Jefferson moves in two directions. If the train is going to Port Jefferson, we say it has positive velocity and if the train is coming to Stony Brook, we say it has negative velocity. However, usually the train takes the same time to travel the same distance, so we can say that the speed is the same for both ways.
We are going to introduce two types of velocities. The average velocity is found by comparing the position of an object at two different moments in time. Say that at $t = t_{1}$ the object is at $x = x_{1}$ and at a later time $t = t_{2}$ the object is at $x = x_{2}$. The average velocity is defined as \[ \bar{v} \equiv \frac{x_{2} - x_{1}}{t_{2} - t_{1}} = \frac{\Delta x}{\Delta t}\] with the displacement $\Delta x \equiv x_{2} - x_{1}$ and the time interval $\Delta t \equiv t_{2} - t_{1}$. This definition can be recognized as the equation for the slope of a line. Indeed, the average velocity is the slope of the secant line that connects the point $(t_{1}, \, x_{1})$ to the point $(t_{2}, \, x_{2})$. It is important to emphasize that this secant line is artificial and it usually does not describe the motion except when the velocity is constant.
On the other hand, the instantaneous velocity is defined at a particular moment in time. The formal definition involves calculus:
\[v(t) \equiv \lim_{\Delta t \rightarrow 0} \frac{x(t + \Delta t) - x(t)}{\Delta t}\]
There is a simple graphical way to understand this expression: the instantaneous velocity is the slope of the tangent line to the position curve at a particular moment in time.
An example: consider a position function of the form
\[x(t) = \frac{1}{2} g (t - t_{0})^{2}\qquad g > 0 \qquad t_{0} > 0\]
This position is a quadratic function of time. Since $g > 0$ we have $x > 0$ for all times. This means that the curve of $x(t)$ only lives in the upper half of the $(t, x)$ plane. The graph is a parabola centered at $t = t_{0}$. We see that for $t < t_{0}$ the position is decreasing in magnitude and the velocity should be negative. When $t = t_{0}$ the object is at the origin. However, at this time the object smoothly turns around for $t > t_{0}$ the magnitude of the position increases. Hence the velocity changed sign. One can check that the velocity function is
\[v(t) = g (t - t_{0})\]
This is a linear function of time. The instantaneous velocity changes with time and it has the same properties we observed earlier: it is negative for $t < t_{0}$, vanishes when $t = t_{0}$ and then becomes positive when $t > t_{0}$.
It is helpful to know how to compare slopes in a graph. The general rule is that lines that increase from left to right have a positive slope. A line that is horizontal is not changing and hence has zero slope. So lines that are close to being horizontal have very small slopes: it is negative if the line is decreasing from left to right or positive if it is increasing. Vertical lines have, in some way, infinite slope: a vertical line is increasing in no time. So lines that are close to the vertical have very large slopes.
Just like position changes with time, velocity also changes with time. The rate of change of the velocity is called the acceleration. Just like for velocity we can talk about the average acceleration:
\[ \bar{a} \equiv \frac{v_{2} - v_{1}}{t_{2} - t_{1}} = \frac{\Delta v}{\Delta t}\]
and instantaneous acceleration:
\[a(t) \equiv \lim_{\Delta t \rightarrow 0} \frac{v(t + \Delta t) - v(t)}{\Delta t}\]
For the example above, since the velocity is linear in time, the slope is constant and the acceleration is constant. One can check that
\[a(t) = g\]
So we have learned that an object with constant acceleration yields a velocity that changes linearly with time and a position that is quadratic in time. This example is very important and we will revisit it later.
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