Section 6.8 Non-Constant Motion
Subsubsection Non-constant Velocity
Suppose you want to find the change in position for a situation where the velocity is a known, non-constant function \(\vec{v}(t)\)
Chop: Start by chopping up a time interval into small pieces. Each piece of time is represented by the expression \(dt\text{,}\) where the \(d\) indicates that the amount of time is infinitesimally small.
Multiply: You want to know about position, but you know about velocity. Although you want \(\Delta \vec{r}\text{,}\) the change in position over a whole time interval, you can start by trying to find an infinitesimal change in position \(d\vec{r}\text{.}\) By rewriting the definition of instantaneous velocity as \(d \vec{r} = \vec{v}(t) dt\text{,}\) you can see that a way to get \(d\vec{r}\) is to multiply \(\vec{v}(t)\) and \(dt\text{.}\)
Add: Last, once you have the infinitesimal \(d\vec{r}\text{,}\) you need to add together every \(d\vec{r}\) over the full times of interest, which can be done with an integral:
\begin{equation*}
\int_{r_i}^{r_f}d \vec{r} = \int_{t_i}^{t_f} \vec{v}(t) dt
\end{equation*}
The left hand side of this equation can be easily evaluated to give
\begin{equation*}
\Delta \vec{r} = \int_{t_i}^{t_f} \vec{v}(t) dt
\end{equation*}
Subsubsection Non-constant Acceleration
Suppose you want to find the change in velocity for a situation where the acceleration is a known, non-constant function \(\vec{a}(t)\)
Chop: Start by chopping up a time interval into small pieces. Each piece of time is represented by the expression \(dt\text{,}\) where the \(d\) indicates that the amount of time is infinitesimally small.
Multiply: You want to know about velocity, but you know about acceleration. Although you want \(\Delta \vec{v}\text{,}\) the change in velocity over a whole time interval, you can start by trying to find an infinitesimal change in velocity \(d\vec{v}\text{.}\) By rewriting the definition of instantaneous acceleration as \(d \vec{v} = \vec{a}(t) dt\text{,}\) you can see that a way to get \(d\vec{v}\) is to multiply \(\vec{a}(t)\) and \(dt\text{.}\)
Add: Last, once you have the infinitesimal \(d\vec{v}\text{,}\) you need to add together every \(d\vec{v}\) over the full times of interest, which can be done with an integral:
\begin{equation*}
\int_{v_i}^{v_f}d \vec{v} = \int_{t_i}^{t_f} \vec{a}(t) dt
\end{equation*}
The left hand side of this equation can be easily evaluated to give
\begin{equation*}
\Delta \vec{v} = \int_{t_i}^{t_f} \vec{a}(t) dt
\end{equation*}
Subsubsection Activities
Activity 6.8.1. Different Displacements.
The following two equations are both reasonable to write down for a displacement vector:
\begin{equation*}
1: \Delta \vec{r} = \int_{t_i}^{t_f}\vec{v}(t)dt
\end{equation*}
\begin{equation*}
2: \Delta \vec{r}= \vec{v}\Delta t
\end{equation*}
What is the difference between them? In what situations would you use the first equation? In what situations would you use the second equation?
