realistic projectile motion. newton’s law e total energy of the moving object, p the power...
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Chapter 2
Realistic projectile motion
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2.1 Frictionless motion with Newton’s law
Newton’s law
E total energy of the moving object, P the power supplied into the system.
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+ O(t2)
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PseudocodeInitialisation: Set values for P, mass m, and time step
t, and total number of time steps, N, initial velocity v0.
Do the actual calculation vi+1= vi + (P/mvi)tti+1= ti + tRepeat for N time steps.
Output the resultsample code 2.1.1, 2.1.2
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Adding friction to the Equation of MotionFrictionless motion is unrealistic as it predicts
velocity shoots to infinity with time.Add in drag force, innocently modeled as
2 C ~ 1, depend on aerodynamics, measured experimentally
B2 CA/2
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The effect of Fdrag: P P – Fdrag v
drag1
ii i
i
P F vv v t
mv
dragdE
P F vdt
2
2
1
2
2
ii
ii i i
i i
C AvP v
C AvPv v t v t t
mmv mv
sample code 2.1.3
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2.2 Projectile motion: The trajectory of a cannon shell
x
y
0
g
Wish to know the position (x,y) and velocity (vx,vy) of the projectile at time t, given initial conditions.
Two second order differential equations.
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2.2 Projectile motion: The trajectory of a cannon shell
Four first order differential equations.
Euler’s method
Eq. 2.16
Eq. 2.15
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2.2 Projectile motion: The trajectory of a cannon shell
Drag force comes in via
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vi = (vx,i2+vy,i
2)1/2
i
i
sample code 2.1.4, 2.1.5
Trajectory of a cannon shell with drag force
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Air density correctionDrag force on a projectile depends on air’s density, which in
turn depends on the altitude.Two types of models for air’s density dependence on
altitude: Isothermal approximation - simple, assume constant
temperature throughout, corresponds to zero heat conduction in the air.
00 1
T
ay K/m105.6 3a
0
2.5 for air;
sea level temperature (in K)T
40 / 1.0 10 my kT mg
•Adiabatic approximation - more realistic, assume poor but non-zero thermal conductivity of air.
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Correction to the drag forceThe drag force w/o correction
corresponds to the drag force at sea-level It has to be replaced by
0drag
*drag
FF
* * *drag, drag drag drag 2
0 0
*drag, 2
0
cos x xx x
y y
v vF F F F B vv
v v
F B vv
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Isothermal approximation:
0 0
1 ,ay
T
* *drag, 2 2 drag, 2
0 0 0
1 ; 1x x x y yay ay
F B vv B vv F B vvT T
Adiabatic approximation:
0 0
expy
y
* *drag, 2 2 drag, 2
0 0 0
exp ; expx x x y yy y
F B vv B vv F B vvy y
K/m105.6 3a
0
2.5 for air;
sea level temperature (in K)T
40 / 1.0 10 my kT mg
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vi = (vx,i2+vy,i
2)1/2
i
i
Trajectory of a cannon shell with drag force, corrected for altitude dependence of air density
0
0
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Curves with thermal and adiabatic correctionModify the existing code to produce the curves as in
Figure 2.5, page 30, Giodano 2nd edition.
sample code 2.1.6, 2.17