Section 6.7 Chop-Multiply-Add: Non-Constant Acceleration
A strategy you will use to solve a variety of physics problems can be thought of as a sequence of steps called Chop-Multiply-Add. You can see these steps outlined in the video above for one example. Below is another example: if you know an object’s acceleration as a function of time and you want to find the change in velocity between some initial and final time.
Chop: You can 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 actual 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*}
Exercises Activities
1.
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?
A*R*C*S 6.7.1. Super Mario’s Acceleration.
Super Mario (mass \(M\)) is initially located at position xi running to the right with constant speed \(v_i\text{.}\) At \(t = 0\text{,}\) Mario finds a Super Star that results in the following force acting on him:
\begin{equation*}
\vec{F}^* = F_o\left(1 - \frac{t}{T}\right)\hat{x}
\end{equation*}
Find the time and position at which Mario has returned to his initial velocity.
1. Analyze and Represent
Representation: sketch a graph of the force as a function of time.
Quantities: Give physical interpretations for \(F_o\) and \(T\)
2. Calculate
First find a symbolic expression for Mario’s velocity as a function of time. Then use your expression to find when Mario’s velocity is equal to \(v_i\text{.}\) Last of all, find Mario’s position.
3. Sensemake
Brainstorm a way to make sense of the equations:
\begin{equation*}
\vec{v}(t) = \left[v_i + \frac{F_o}{M} \left(t - \frac{t^2}{2T}\right)\right] \hat{x}
\end{equation*}
\begin{equation*}
t_f = 2T
\end{equation*}
\begin{equation*}
\vec{x}(t) = \left[x_i + v_i t + \frac{F_o}{M} \left(\frac{t^2}{2} - \frac{t^3}{6T}\right)\right] \hat{x}
\end{equation*}
\begin{equation*}
\vec{x}_f = \left[x_i + 2v_i T + \frac{2}{3} \frac{F_o}{M} T^2\right] \hat{x}
\end{equation*}
