Kinematics 1

New Maple Functions

• solve  This function solves equations and systems of equations.  For an example that displays some of the available Greek alphabet, assign the following quadratic expression to the variable quad_exp.

  > quad_exp := alpha*x^2 + beta*x + gamma;

  

  Now use solve to find the roots of the quadratic equation quad_exp = 0,

  > solve( quad_exp = 0, x );

  

  Hopefully these solutions look familiar.

  You may use the set operator ( { } ) to create a system of equations, then use solve to find the solutions.

  > eqns := { 2*x + 7*y = 15, -8*x + y = -3 };

  

  > solve( eqns, { x, y } );

  

  Note that solve does not assign the found solutions to the variables.

  > x, y;

  

  For more details check out ?solve and ?set.

• subs  This function substitutes variables into an expression and then evaluates the expression.

  > subs( x = 5, quad_exp );

  

  > subs( { x = 18/29, y = 57/29 }, eqns );

  

  You might sometimes use subs to solve for variables, but more often you will use it to check Maple's calculations, as in the above example.  Check out ?subs if you want more details.

• lhs / rhs  Use these functions to manipulate or represent the left or right hand side of an equation.  For example, remembering that Maple makes no judgement on the validity of an equation,

  > gender_equality := 2*women = 3*men;

  

  > lhs( gender_equality );

  

  > rhs( gender_equality );

  

  > lhs( gender_equality ) = rhs( gender_equality );

  

  Look at ?lhs or ?rhs for more details.

Top

The Fall of the Rubber Duck

You work as a camera person for a film crew.  Currently the film crew is shooting an action thriller.  The plot of this culturally enriching film is pretty simple.  An emotional circus clown is unhappy with the quality of the makeup that his boss provides.  The irritation builds and finally the clown flips, kidnapping one of the boss' prized waterfowls.  The rest of the movie involves one big chase scene after another, with some concocted hero type trying to catch the clown and retrieve the little duck.  As is typical with these movies, at one point during the chase, with the hero on his tail, the clown logically runs to the edge of a high cliff.  Caught and desperate for ultimate revenge, he drops the little duck off the edge of the cliff.  Now this is a big deal for this little duck because the control freak boss clips the wings of all the birds in his collection!  This is a low budget film and the production company only purchased one simulation rubber duck.  Hauling this rubber duck up and down a big cliff would be time consuming and expensive. 

In order to shoot the rubber duck's fall in one take, you need to use your knowledge of physics to make some predictions about the behavior of the fall of the rubber duck.  Some things you know: the cliff is 500 m high, the clown holds the rubber duck over the side of the cliff and simply lets go of it, the duck will experience a constant rate of acceleration of 10 m/s^2, and this is beginning physics, so as unlikely as it may seem, the atmosphere contains no air.  Some potentially useful pieces of information that you want to figure out: the total time of the rubber duck's fall, its velocity at the moment of ultimate doom, its average velocity during the fall, and perhaps some graphs to show how the duck's position and velocity change during the fall.

Solution

As mentioned in Maple Think, on most computers you may copy and paste the bold red Maple command lines from your browser window to your Maple worksheet.  Also, do not forget that in your Maple worksheet you may place the cursor on a Maple word, and then select "Help on..." from the Help menu to view a related help page.

To avoid confusion and safely reuse variables it's a good idea to always start a problem solution with

> restart;

Always start with a free-hand sketch of a situation.   Make a sketch with ground level at height 0 m, and the top of the cliff at height 500 m.  Note that you just imposed a coordinate system on the physical situation.

> y_pos := y[f] = y[i] + v[i]*t + (1/2)*a*t^2;

> y[i] := 500;

> y[f] := 0;

> v[i] := 0;

> a := -10;

Now that values are assigned to the other variables, the equation y_pos contains only one unknown, t.

> y_pos;     Note that full evaluation takes place in this command.

> solve( y_pos, t );

Always check your results.  It's an important and often neglected part of problem solving, but hopefully a CAS like Maple will help you develop the habit.  Checks of your work with a CAS are typically faster than checks done by hand.  

> subs( t = 10, y_pos );

Thus, the total time for the rubber duck's fall is 10 s.

> vel := v[f] = v[i] + a*t;

> subs( t = 10, vel );

And as a check,

> subs( {t = 10, v[f] = -100}, vel );

And using the definition of average velocity,

> v[avg] := ( y[f] - y[i] )/10;       s

> displacement = v[avg]*10;

This is the displacement vector of the rubber duck's fall along the y-axis as stated in the problem statement, and confirms the accuracy of our calculations.

Now for the graphs of how the rubber duck's position and velocity change during the fall.  Try the following.

> y = rhs( y_pos );

Now select "2D" from "Plot" in the Insert menu.  This should create a live plot frame on your screen.  Now select the Maple output of the above command by dragging the mouse over it.  Now place the mouse over the selected region and press the left mouse button without releasing it.  Now drag the mouse over to the center of the live plot frame and release the button.  The selected Maple output should disappear, and a graph of the equation should appear in the live plot frame. 

Now you need to scale the axes according to the parameters of the problem.  Place the mouse over the live plot frame and click the right mouse button.  Select "Ranges..." from "Axes" and then click the radio buttons in the resulting dialog box to change the scaling on the axes.  Change the horizontal axis to 0 - 10, and change the vertical axis to 500 - 0.  The result should look very similar to the graph below.

Clicking in the live plot frame creates a plot toolbar with a variety of buttons.  Click and unclick each of them to see how to quickly alter the look of a graph.  Not all of them produce a noticeable change in this particular graph.   Also note that left clicks in the plot frame cause the coordinates of the mouse to appear at the left end of the plot toolbar.

Plot the changes in the rubber duck's velocity in a similar manner.

> v = rhs( vel );

The Kinematics 1 Problem Set includes problems similar to the one above.  Complete Maple solutions are provided.

Kinematics 1 Problem Set     Top     Kinematics 2

Tutorial Index      Maple Index         How to...     Problem Set Index

Please send me any polite comments, suggestions, or corrections.