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*************************************
What is your assignment?
1) Execute the Sample Problem #1 (Constant Velocity), and answer the questions at the end.
2) I have duplicated Sample Problem #1.
Your mission (should you choose to accept) is to
modify this problem so that it is a problem of CONSTANT ACCELERATION. Use a=10.
:[font = section; inactive; noPageBreak; preserveAspect; startGroup]
*****************************
1)Sample Problem: Constant Velocity
:[font = subsection; inactive; noPageBreak; preserveAspect]
*************************************
Part A) Position
:[font = text; inactive; noPageBreak; preserveAspect]
First, let's start by clearing all the variables.
:[font = input; noPageBreak; preserveAspect]
Clear["Global`*"]
:[font = text; inactive; noPageBreak; preserveAspect]
Let's start by defining a function for the postition x.
:[font = input; noPageBreak; preserveAspect; startGroup]
x[t_]= x0 + v0 t
:[font = output; output; inactive; preserveAspect; endGroup]
t*v0 + x0
;[o]
t v0 + x0
:[font = text; inactive; noPageBreak; preserveAspect]
Note how we use the notation t_ with the underscore "_" to tell Mathematica that t is a dummy variable. To see what I mean, try:
;[s]
3:0,0;65,1;76,0;131,-1;
2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0;
:[font = input; noPageBreak; preserveAspect; startGroup]
x[2]
:[font = output; output; inactive; preserveAspect; endGroup]
2*v0 + x0
;[o]
2 v0 + x0
:[font = text; inactive; noPageBreak; preserveAspect]
Now we pick values for x0 and v0.
:[font = input; noPageBreak; preserveAspect]
v0=5;
x0=0;
:[font = text; inactive; noPageBreak; preserveAspect]
Now, x[t] at t=2 is a number.
:[font = input; noPageBreak; preserveAspect; startGroup]
x[2]
:[font = output; output; inactive; preserveAspect; endGroup]
10
;[o]
10
:[font = text; inactive; noPageBreak; preserveAspect]
We can now make a plot of x[t] vs. t for times from t=0 to t=5. Note how I turn on the GridLines.
:[font = input; noPageBreak; preserveAspect; startGroup]
Plot[ x[t] ,{t,0,5}, GridLines->Automatic];
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We can find the total distance traveled in 2 ways.
First, we can just read the answer off the graph.
Second, we can evaluate
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x[5]-x[0]
:[font = output; output; inactive; preserveAspect; endGroup]
25
;[o]
25
:[font = text; inactive; noPageBreak; preserveAspect]
Both methods give 25 meters.
:[font = subsection; inactive; noPageBreak; preserveAspect]
*************************************
Part B) Velocity
:[font = text; inactive; noPageBreak; preserveAspect]
We can use the fact that v = dx/dt. Let's make a function v[t] using this definition. We start by clearing the previous definitions of x0 and v0 so we can see the formula in terms of symbols and not numbers.
:[font = input; noPageBreak; preserveAspect]
Clear[x0,v0]
:[font = input; noPageBreak; preserveAspect; startGroup]
v[t_]=D[x[t],t]
:[font = output; output; inactive; preserveAspect; endGroup]
v0
;[o]
v0
:[font = text; inactive; noPageBreak; preserveAspect]
The answer is that v[t]=v0, a constant.
(We already knew this since this was a constant velocity problem, but we used Mathematica to double check.)
Now let us set x0 and v0 to our previous values.
;[s]
3:0,0;119,1;130,0;199,-1;
2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0;
:[font = input; noPageBreak; preserveAspect]
v0=5;
x0=0;
:[font = text; inactive; noPageBreak; preserveAspect]
and plot the velocity.
Note, I have added some extra lines to the plot command to make it look nice.
(If you like, experiment by taking out certain parts to see what each does.)
:[font = input; noPageBreak; preserveAspect; startGroup]
Plot[ v[t] ,{t,0,5}
,GridLines->Automatic
,PlotRange->{0,6}
,PlotStyle->{Thickness[0.020]}];
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The velocity curve is supposed to be the slope of the distance (x[t]) curve. Is this true?
Show this by marking on the plot of x[t] and v[t]
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The distance traveled is supposed to be the area under the velocity curve. Is this true?
How much distance is represented by each square grid on the graph???
By counting the grid squares under the curve (estimat this) comput the distance traveled from t=0 to t=5??? From t=2 to t=4???
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Calculate directly the distance traveled from t=0 to t=5??? From t=2 to t=4??? Compare with what you got by counting the grids.
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2) Problem: Constant Acceleration
Things have to be added!!!
This is not the complete answer!!!
;[s]
2:0,0;65,1;127,-1;
2:1,19,14,Times,1,18,0,0,0;1,16,12,Times,1,14,0,0,0;
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*************************************
Part A) Position
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First, let's start by clearing all the variables.
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Clear["Global`*"]
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Let's start by defining a function for the postition x.
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(*** Not Complete. You fix it ***)
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x[t_]= x0 + v0 t
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Note how we use the notation t_ with the underscore "_" to tell Mathematica that t is a dummy variable. To see what I mean, try:
;[s]
3:0,0;65,1;76,0;131,-1;
2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0;
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x[2]
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Now we pick values for x0 , v0, and a.
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v0=5;
x0=0;
a=10;
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Now, x[t] at t=2 is a number.
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x[2]
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We can now make a plot of x[t] vs. t for times from t=0 to t=5. Note how I turn on the GridLines.
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Plot[ x[t] ,{t,0,5}, GridLines->Automatic];
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We can find the total distance traveled in 2 ways.
First, we can just read the answer off the graph.
Second, we can evaluate
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x[5]-x[0]
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Both methods give ??? meters.
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Part B) Velocity
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We can use the fact that v = dx/dt. Let's make a function v[t] using this definition.
We start by clearing the previous definitions of x0, v0, and a so we can see the formula in terms of symbols and not numbers.
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Clear[x0,v0,a]
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v[t_]=D[x[t],t]
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Now let us set x0, v0, and a to our previous values.
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v0=5;
x0=0;
a=10;
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and plot the velocity.
Note, I have added some extra lines to the plot command to make it look nice.
(If you like, experiment by taking out certain parts to see what each does.)
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Plot[ v[t] ,{t,0,5}
,GridLines->Automatic
,PlotStyle->{Thickness[0.020]}];
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The velocity curve is supposed to be the slope of the distance (x[t]) curve. Is this true?
Show this by marking on the plot of x[t] and v[t]
:[font = subsubsection; inactive; noPageBreak; preserveAspect]
The distance traveled is supposed to be the area under the velocity curve. Is this true?
How much distance is represented by each square grid on the graph???
By counting the grid squares under the curve (estimat this) comput the distance traveled from t=0 to t=5??? From t=2 to t=4???
:[font = subsubsection; inactive; noPageBreak; preserveAspect; endGroup]
Calculate directly the distance traveled from t=0 to t=5??? From t=2 to t=4??? Compare with what you got by counting the grids.
^*)