Section 5.1 One-Dimensional Motion with Constant Acceleration
“-----”―------
In this section, we will be focusing on describing the kinematics objects when there is a constant acceleration. Before getting started, we are going to introduce some basic notation.
- \(a\text{:}\) The constant acceleration.
- \(v = v(t)\text{:}\) Velocity as a function of time. This notation means that we will sometimes use a \(v\) by itself, but you need to remember that this changes with time. If we need to explicitly indicate the time, we will use the second notation with \(t\) being the specific value.
- \(v_0 = v(0)\text{:}\) Initial velocity. Notice that this notation means that we are using the velocity at time \(t = 0\) as the value of \(v_0\text{.}\)
- \(s = s(t)\text{:}\) Position as a function of time. This follows the same notation rules as velocity.
- \(s_0 = s(0)\text{:}\) Initial position.
The Constant Acceleration Equations.
Here are the equations for position and velocity when the acceleration is constant:
\begin{align*}
s(t) \amp = \frac{1}{2} at^2 + v_0 t + s_0\\
v(t) \amp = at + v_0
\end{align*}
These equations create the mathematical curves of a parabola and a line. This means that the underlying mathematics of these motions are not too complicated. In a regular physics course, we would spend a lot of time solving equations, but for this course our emphasis will be on building intuition that helps us to understand the motion instead of doing calculations.
In the special case that we’re looking at the acceleration due to earth gravity, we use \(a = -g\text{,}\) where \(g = 9.8 \frac{\text{m}}{\text{s}^2}\text{.}\) Then the equations would look like this:
\begin{align*}
s(t) \amp = -4.9 t^2 + v_0 t + s_0\\
v(t) \amp = -9.8t + v_0
\end{align*}
It is important to remember that these equations are for movement in just one dimension. If we are working in two dimensions, each dimension will have its own equations. (We will explore that scenario in the next section.)
Parabolas.
We need to spend a little bit of time talking about the graph of a parabola because it is fundamentally important to kinematics. We will use the graph of an object being launched vertically from the ground as our basic model for discussing the parabola’s central features.
The first observation of the parabola is the shape. Intuitively, it makes sense that if we launch something upward, it will eventually turn downward because of gravity. And the graph of the parabola matches this behavior. This shape is due to the negative sign on the \(t^2\) term. If that term were positive, we would have an upward-opening parabola, and while this is perfectly meaningful mathematically, we do not have a lot of physics contexts where the parabola faces this way.
The parabola has a point where the motion changes from going up to going down. That point is called the vertex of the parabola. This point also marks the axis of symmetry of the parabola, meaning that the shape of the parabola moving away from this point is the same in both directions. This means that if you were to watch an object in motion, you would not be able to tell whether time was going forward or backward based on the motion of the object alone. It looks exactly the same going up as it does going down with the time reversed.
If we were to drop an object from rest (without any initial velocity), we would get a half-parabola shape that starts from the vertex and falls down. The parabola from here has an interesting pattern. Suppose we picked a certain amount of time (such as 2 seconds) and measured how far it fell during that time. If we were to let it fall for twice the amount of time (4 seconds), it will have fallen four times the initial measurement. If we were to let it fall for three times the amount of time (6 seconds), it will have fallen nine times the initial measurement. This is the nature of the shape of a parabola.
