4  Lab 4: Rate of Change

Objective

The objective of this lab is to deepen understanding of the concepts of average and instantaneous velocity, as well as their related measures such as speed and acceleration. This will be achieved through both theoretical definitions and practical applications, including symbolic computation with Maxima and manual calculations. Specifically, students will:

  1. Review Key Definitions:
    • Understand the definitions of average velocity and instantaneous velocity.
    • Learn the relationships between velocity, speed, and acceleration for an object moving along a line with position \(s = f(t)\).
  2. Apply Definitions to a Real-World Problem:
    • Analyze the motion of a stone thrown vertically upward, including the derivation of its velocity and acceleration functions.
    • Determine the highest point reached by the stone and the velocity at which it strikes the river.
  3. Utilize Maxima for Symbolic Computation:
    • Use Maxima to solve the given problem, deriving velocity and acceleration functions, finding the highest point, and calculating the impact velocity.
    • Learn to define functions, perform differentiation, solve equations, and evaluate expressions within Maxima.
  4. Practice Problem-Solving Skills:
    • Solve the given problem by hand to reinforce understanding and manual calculation skills.
    • Document the handwritten solution and integrate it into the Maxima-generated document.
  5. Extend Learning through Additional Exercises:
    • Select three similar problems from the homework or textbook.
    • Solve these problems using both Maxima and manual methods, and compile the solutions, including handwritten work, in the final document.

By the end of this lab, students will have a solid grasp of the theoretical concepts, practical application, and computational tools necessary to analyze motion along a line, enhancing both their analytical and technical skills.

4.1 Definitions for average velocity and instantaneous velocity

4.1.1 Average velocity

\[ v_{avg} = \frac{f(a+\Delta t)-f(a)}{\Delta t}\]

4.1.2 Instantaneous velocity

\[ v(a) = \lim_{\Delta t \rightarrow 0}\frac{f(a+\Delta t)-f(a)}{\Delta t}=f'(a)\]

4.2 Definitions for velocity, speed and acceleration

Suppose an object moves along a line with position \(s=f(t)\). Then

\[\text{velocity at }t:\qquad v =\frac{ds}{dt}=f'(t)\] \[\text{speed at }t:\qquad |v| =|f'(t)|\] \[\text{acceleration at }t:\qquad a =\frac{dv}{dt}=\frac{d^2s}{dt^2}=f''(t)\]

4.3 Example

Suppose a stone is thrown vertically upward with an initial velocity of \(64\,ft/s\) from a bridge \(96\,ft\) above the river. By Newton’s laws of motion, the position of the stone (measured as the height above the river) after \(t\) seconds is \[s(t)=-16t^2+64t+96\] where \(s=0\) is the level of the river.

  1. Find the velocity and acceleration functions.
  2. What is the highest point above the river reached by the stone?
  3. With what velocity will the stone strike the river?

4.3.1 Solution using maxima

Sol a:

(%i1) s(t):= -16*t^2+64*t+96;

\[\mathtt{(\textit{%o}_{1})}\quad s\left(t\right):=-16\,t^2+64\,t+96\]

(%i2) define(v(t),diff(s(t),t,1));

\[\mathtt{(\textit{%o}_{2})}\quad v\left(t\right):=64-32\,t\]

(%i3) define(a(t),diff(s(t),t,2));

\[\mathtt{(\textit{%o}_{3})}\quad a\left(t\right):=-32\]

Sol b:

(%i4) solb: solve(v(t)=0,t);

\[\mathtt{(\textit{%o}_{4})}\quad \left[ t=2 \right] \]

(%i5) solb[1];

\[\mathtt{(\textit{%o}_{5})}\quad t=2\]

(%i6) rhs(%);

\[\mathtt{(\textit{%o}_{6})}\quad 2\]

(%i7) s(%);

\[\mathtt{(\textit{%o}_{7})}\quad 160\]

Sol c:

(%i8) solc: solve(s(t)=0,t);

\[\mathtt{(\textit{%o}_{8})}\quad \left[ t=2-\sqrt{10} , t=\sqrt{10}+2 \right] \]

(%i9) solc[2];

\[\mathtt{(\textit{%o}_{9})}\quad t=\sqrt{10}+2\]

(%i10) rhs(%);

\[\mathtt{(\textit{%o}_{10})}\quad \sqrt{10}+2\]

(%i11) v(%);

\[\mathtt{(\textit{%o}_{11})}\quad 64-32\,\left(\sqrt{10}+2\right)\]

(%i12) float(%);

\[\mathtt{(\textit{%o}_{12})}\quad -101.19288512538816\]

(%i13) round(v(rhs(solc[2])));

\[\mathtt{(\textit{%o}_{13})}\quad -101\]

4.4 Solve same problem by hand

  • In a piece of paper, solve same problem by hand.
  • Take a picture and save it in the same place you have your maxima file.
  • Include that picture to your document using maxima:

Go to: Cell -> Insert Image

4.5 From the homework 3.6 in mymathlab (or the book)

  • Find 3 similar problems and repeat previous steps:
    • Solve it using maxima
    • Solve it by hand and attach the handwritten solution to your maxima document.

4.6 For each of the previous sections:

  • take pictures of your handwritten work, submit them to the dropbox.
  • Link each picture to the corresponding section of your wxmaxima document1 (Cell -> Insert Image).
  • be sure your code run correctly by pressing Shift+Enter in each code chunk.
  • be sure the whole document display correctly, by running Ctrl+R.
  • Submit on the corresponding Dropbox folder, with the corresponding name (FINAL_Lab4_Calc1_YOURLASTNAME_mmddyy.wxmx.)

  1. pictures need to be in the same folder of your .wxmx file↩︎