Name:                 Jim Hays

Title:                   Trajectories - They're not what you think!

Description:        This lesson is intended for a high school physics class demonstration/lab after the students have been exposed to projectile motion in the usual treatment.

Trajectories
They are not what you think!

Looking at the trajectile from the usual standpoint, we would ignore any air resistance, wind, etc. and consider that the trajectile travels along a parabolic path.  Consider the situation below.  The initial velocity of the object is 50 m/s.  The acceleration due to gravity is -9.8 m/s¤.  The inital angle is 45 degrees.  Enable the cells below to view the path of the object.

Clear[fx,fy,st,x,y,v0,vx,vy,th,th2,fr,t,gr]
st = 0.1
gr = -9.8
th = 45 Degree
v0 = 50;

vx = N[v0  Cos[th]]
vy = N[v0  Sin[th]]
fx[t_] = N[(vx ) t]
fy[t_] = N[(vy )t  + 1/2 gr t^2] 
tfinal = FindRoot[fy[t],{t,100}][[1,2]]
Stuff = Table[{fx[t],fy[t]},{t,0,tfinal,st}]
ListPlot[Stuff,AspectRatio -> Automatic,PlotStyle -> RGBColor[1,0,0]];

How far down range did the object travel?

A little over 250m.

What was the maximum height of the object?

About 60m.

This is the treatment that most high school (and early college) textbooks take.  They ignore any air resistance and/or friction, etc. that might have an affect on the path of the object.  However, if we think about the situation a little bit, we know that air resistance, friction, etc. will have an affect on the path.  Let's look at this same situation again.  This time, let's consider that the air resistance, etc. has the affect of slowing the object down by a factor of 0.01 each time period.  (Constant decelleration....)  This decelleration will not only affect the vertical component of the velocity vectors, but the horizontal components as well.

Let's look at the path of a trajectile that has been launched with an initial velocity of v0, an angle of th, constant decelleration of fr, and acceleration due to gravity of gr.  Enable the cell below now, please.

Clear[fx,fy,st,x,y,v0,vx,vy,th,th2,fr,t,gr]
st = 0.1
gr = -9.8
th = 45 Degree
v0 = 50
fr = 0.01;

Now enable the cell below.  This is the path that a trajectile that has an intial velocity of 50, and angle of 45 degrees, a coefficient of friction of 0.1, and an acceleration due to gravity of -9.8 would take.

vx = N[v0  Cos[th]]
vy = N[v0  Sin[th]]
fx[t_] = N[(vx (1 - fr) ^ t) t]
fy[t_] = N[(vy (1 - fr) ^ t) t + 1/2 gr t^2] 
tfinal = FindRoot[fy[t],{t,100}][[1,2]]
Stuff = Table[{fx[t],fy[t]},{t,0,tfinal,st}]
ListPlot[Stuff,AspectRatio -> Automatic,PlotStyle -> RGBColor[1,0,0]];

Approximately how far does the projectile travel down range?

About 220 meters.

Approximately how high does the projectile go?

About 60 meters.

Why does this object not go as far down range or as high as the object in the previous example?  It has the same initial velocity vector.

Now, change the values in the cell below to the following:
v0 = 100
th = 45 Degree
gr = -9.8
fr = 0.1
and enable the cell.

Clear[fx,fy,st,x,y,v0,vx,vy,th,th2,fr,t,gr]
st = 0.1
gr = -9.8
th = 45 Degree
v0 = 100
fr = 0.1;

Now enable the cell below to see the path.

vx = N[v0  Cos[th]]
vy = N[v0  Sin[th]]
fx[t_] = N[(vx (1 - fr) ^ t) t]
fy[t_] = N[(vy (1 - fr) ^ t) t + 1/2 gr t^2] 
tfinal = FindRoot[fy[t],{t,100}][[1,2]]
Stuff = Table[{fx[t],fy[t]},{t,0,tfinal,st}]
ListPlot[Stuff,AspectRatio -> Automatic,PlotStyle -> RGBColor[1,0,0]];

Now, change the values in the cell below several times and enable the two cells.  Experiment.....  Change each of the values..  v0, th, fr, and lastly gr.  Be prepared to explain how each of these values affect the trajectory.

Once you have experimented with last set of cells, go back to the first two sets of cells.  Change the values in each set of cells, leaving everything the same except for fr.  Be prepared to tell what happens as this decelleration changes.

Questions to ponder.....

What angle will give tha maximum horizontal distance at a given initial velocity?  Is this what you expected?

What happens as the inital velocity changes?

What happens as gravity changes?

What happens as the decelleration due to air resistance, etc. changes?