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Question: plotting without values

Question:plotting without values

Christopher2222 5800
plot value
October 11 2012
0

Can we plot without values to variables that we know are constant or do we always need to assign some kind of a value?  

As an example lets just go back to a first/second year classical mechanics course.  We'll just use the simple case of a falling projectile.  The object has a velocity of v0, with a gravitational force mg and resistive force kmv.  Using Newtons 2nd law of motion we can set up the equation.

Anyways as always maple easily handles the equations for the general case ...


eq1 := m*(diff(z(t), t, t)) = -m*g-k*m*(diff(z(t), t))

m*(diff(diff(z(t), t), t)) = -m*g-k*m*(diff(z(t), t))

 (1)

ic := z(0) = h, (D(z))(0) = v0

z(0) = h, (D(z))(0) = v0

 (2)

eq2 := dsolve({eq1, ic}, z(t))

z(t) = -exp(-k*t)*(v0*k+g)/k^2-g*t/k+(h*k^2+v0*k+g)/k^2

 (3)

Now this is not exactly what the book gives.  The book gives:

 

eq3 := z = h-g*t/k+(v0*k+g)*(1-exp(-k*t))/k^2

z = h-g*t/k+(v0*k+g)*(1-exp(-k*t))/k^2

 (4)

However, they are in fact the same

rhs(eq2)

-exp(-k*t)*(v0*k+g)/k^2-g*t/k+(h*k^2+v0*k+g)/k^2

 (5)

rhs(eq3)

h-g*t/k+(v0*k+g)*(1-exp(-k*t))/k^2

 (6)

is(rhs(eq2) = rhs(eq3))

true

 (7)

It would take a little work to get the equation Maple presented into the form that was in the textbook presented by the author.

 

collect(eq2, 1/k^2)

z(t) = h+(-exp(-k*t)*v0-g*t+v0)/k+(-g*exp(-k*t)+g)/k^2

 (8)

Perhaps there exists a simple way, any attempts are welcome.

 

In the text there is a general plot, I was hoping could easily be done from the general equation in Maple.  

``

The terminal velocity vt shown as the line -g/k in the graph.

 

So, to find our terminal velocity I am recalculating our 2nd ode with velocities and setting the acceleration=0

``

eq4 := m*(diff(v(t), t)) = -m*g-k*m*v(t)

m*(diff(v(t), t)) = -m*g-k*m*v(t)

 (9)

solve(subs(diff(v(t), t) = 0, eq4), v(t))

-g/k

 (10)

 

It is not so simple to plot the graph in Maple without the values is it?


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