1. 3. Motion in the presence of the air resistance
Air resistance (drag force)
Linear term from viscosity and dominated at laminar flow.
Quadratic term from having to accelerate fluid, dominated at turbulent flow.
For a sphere with diameter D:
- fluid viscosity
- fluid density
- Reynolds number
For air:
1.7x10-5 Ns/m2
1.6x10-4 Ns/m2
1.29 kg/m3
0.25 Ns2/m4
(1.6x103 s/m2)Dv

2. Examples:
1) Baseball
Conclusion: the linear term can be neglected!
2) Drop of rain
Conclusion: the two term are comparable
and neither can be neglected!
3) Millikan oil drop
Conclusion: the quadratic term can be neglected!

3. Linear air resistance in the presence of gravity (velocity)
Coordinate y is directed vertically down.
Solution for x-component:
Solution for y-component:
Comments about the sign of:

4. Linear air resistance in the presence of gravity (position)
Coordinate y is directed vertically upward.
Limits:
Coordinate x has a limit!
Trajectory:

5. Linear air resistance in the presence of gravity (trajectory)
Vertical asymptote
Range is the solution of the transcendental equation:

6. Range (dimensionless variables)

7. Quadratic air resistance (Horizontal motion)
x is increasing with time!

8. Matlab cod %Quadratic air resistance (horizontal motion)
t=0:0.05:3; %time
v=1./(1+t); %velocity
x=log(1+t); %position
figure(1)
plot(t,v)
xlabel('t/\tau')
ylabel('v/v_0')
title('Velocity as a function of time')
figure(2)
plot(t,x)
ylabel('(x-x_0)/v_0\tau')

9. Quadratic air resistance (Vertical motion)
Coordinate y is directed vertically down.

10. Matlab cod
%Quadratic air resistance (vertical motion)
vy=tanh(t);
y=log(cosh(t));
figure(3)
plot(t,vy)
xlabel('gt/v_t_e_r')
ylabel('v/v_t_e_r')
title('Velocity as a function of time')
figure(4)
plot(t,y)
ylabel('g(y-y_0)/v^2_t_e_r')
title('Position as a function of time')

11. Quadratic air resistance (Vertical motion; absolute units)

12. %Quadratic air resistance (vertical motion); absolute units
g=9.8;
vter=35;
k1=g/vter;
k2=vter^2/g;
ta=t/k1;
va=vter*tanh(t);
ya=k2*log(cosh(t));
figure(5)
plot(ta,va,ta,g*ta)
xlabel('time [s]')
ylabel('velocity [m/s]')
title('Velocity as a function of time')
axis([0,12,0,40])
legend('with air resistance','in vacuum')
figure(6)
plot(ta,ya,ta,g*ta.^2/2)
ylabel('position [m]')
title('Position as a function of time')
axis([0,12,0,300])
Matlab cod

13. Quadratic air resistance (Horizontal and vertical motion)
Coordinate y is directed vertically upward.