A*R*C*S 6.10.1. 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.
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*}
