Chapter 2
Realistic projectile motion
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.
+ O(Dt2)
Pseudocode
 Initialisation: Set values for P, mass m, and time step
Dt, and total number of time steps, N, initial velocity
v0.
 Do the actual calculation
 vi+1= vi + (P/mvi)Dt
 ti+1= ti + Dt
 Repeat for N time steps.
 Output the result
sample code 2.1.1, 2.1.2
Adding friction to the Equation of Motion
 Frictionless motion is unrealistic as it predicts
velocity shoots to infinity with time.
 Add in drag force, innocently modeled as
B2  CrA/2
2
C ~ 1, depend on aerodynamics, measured experimentally
 The effect of Fdrag: P  P – Fdrag v
dE
 P  Fdrag v
dt
vi 1  vi
P  Fdrag vi 


Dt
mvi

C r Avi 2 
vi 
 P 

2
C r Avi 2
P


vi 1  vi 
Dt  vi 
Dt 
Dt
2m
mvi
mvi
sample code 2.1.3
2.2 Projectile motion: The trajectory of a
cannon shell
Two second order differential
equations.
Wish to know the position
(x,y) and velocity (vx,vy) of the
projectile at time t, given initial
conditions.
y
g
0
x
2.2 Projectile motion: The trajectory of a
cannon shell
Four first order differential
equations.
Euler’s
method
Eq. 2.15
Eq. 2.16
2.2 Projectile motion: The trajectory of a
cannon shell
Drag force comes in via
Trajectory of a cannon shell with drag
force
vi = (vx,i2+vy,i2)1/2
i
i
sample code 2.1.4, 2.1.5
Air density correction
 Drag 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.
y0  kT / mg  1.0 104 m
•Adiabatic approximation - more realistic, assume poor
but non-zero thermal conductivity of air.

 ay 
r  r 0 1  
 T0 
  2.5 for air; a  6.5  103 K/m
T0 sea level temperature (in K)
Correction to the drag force
 The drag force w/o correction
corresponds to the drag force at sea-level
 It has to be replaced by
r 
*
Fdrag
 Fdrag   
 r0 
*
*
*
Fdrag,
x  Fdrag cos   Fdrag
 r 
*
Fdrag,


B
vv


y
2 y 
r
0
 
 r  v
 r 
vx
 Fdrag     x   B2 vv x   
v
 r0  v
 r0 
Isothermal approximation:
 y 
r
 exp   
r0
 y0 
 r 
 y  *
 y 
*
Fdrag,


B
vv



B
vv
exp

;
F


B
vv
exp


 drag,y
 
x
2 x 
2 x
2 y
r
y
 0
 0
 y0 
y0  kT / mg  1.0 104 m
Adiabatic approximation:

r  ay 
 1   ,
r0  T0 

*
Fdrag,
x

 r 
 ay 
 ay 
*
  B2vvx      B2vv x 1   ; Fdrag,


B
vv

y
2 y 1 
r
T
0 
 0

 T0 
  2.5 for air; a  6.5  103 K/m
T0 sea level temperature (in K)
Trajectory of a cannon shell with drag force,
corrected for altitude dependence of air density
vi = (vx,i2+vy,i2)1/2
i
 r 


 r0 
i
 r 


r
 0
Curves with thermal and adiabatic
correction
 Modify the existing code to produce the curves as in
Figure 2.5, page 30, Giodano 2nd edition.
sample code 2.1.6, 2.17