Projectile motion + Air resistance equation?

In summary: Unfortunately, in projectile motion there are several complexities which must be taken into account in order to calculate the range of a projectile. First, air resistance is not a constant, it changes as a function of projectile velocity. This means that the range of the projectile will change depending on the velocity chosen. Second, force of air resistance is not a constant, it also changes as a function of velocity. This means that the range of the projectile will also change depending on the velocity chosen. Third, the range of a projectile can also be calculated using a kinematic equation. However, this equation is not always a good fit for the data. In order to solve the kinematic equation, you need to
  • #1
Thaer
1
0
Take a look at this little web-progarm
http://galileo.phys.virginia.edu/classes/109N/more_stuff/Applets/ProjectileMotion/jarapplet.html

I know that it uses the following equation to calculate the range of the projectile without air resistance
R = (v^2 sin2(theta))/g
R - Range
v - intial velocity

But I have no clue how it calculates range of the projectile with air resistance.
Does anyone know what equation(s) the program uses to calculate the range of the projectile with air resistance (air resistance enabled in the program)
 
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  • #2
You just need to know the direction(s) of the resistance force.Assuming the projectile takes place in x-y plane and the air resistance force is given by:

[itex]F_r=F_x i + F_y j[/itex]

You need not remember the formulae, just apply Newton 'kinematic equations' in x direction and y direction seperately , in this case taking into consideration the 'acceleration/decelleration due to air resistance force'

BJ
 
  • #3
Thaer said:
But I have no clue how it calculates range of the projectile with air resistance.
Does anyone know what equation(s) the program uses to calculate the range of the projectile with air resistance (air resistance enabled in the program)

Air resistance does not fit into a simple equation. Force of air resistance is not a constant, it changes as a function of projectile velocity. You can assume

F drag = C * v or F drag = C * v^2

C is the coefficient of drag for an object. It depends on numerous variables.
Experiments show that the later relationship is a better curve fit. You can bet that the equation(s) they used are formulated on a combination of theoretical and experimental results. You might consider going to the source and asking them.
 
  • #4
As Pete explained, air resistance is a nonlinear function of velocity, so it is time varying. The only way to solve the differential equations of the movement is numerically. You attribute an initial value to the velocity, calculate the drag and from that the accelerations in the horizontal and vertical directions. You give a small time increment and calculate the new value of the velocity. Work iteratively until you find that the height is zero. The horizontal distance at that instant is the range.
 
  • #5
The model (differential equation) of the projectile motion will depend on the relationship between the resistive force and the velocity of the projectile that is chosen. Will it be f=k v, f=k v_squared? Wikipedia has some relevant comments in this area and the site at math.fullerton.edu/mathews/n2003/ProjectileMotionMod/ (continues beyond my page) has Mathematica programs which contain the differential equations for the f=k v case and the f=k v_squared case.
 
  • #6
Hi Thaer,

The web-program you've mentioned very likely uses the air resistance as proportional to v or v^2, as has been mentioned by others. For the kv, you get a nice analytical solution in 2-d. Not so in the kv^2 case. But when the object is thrown vertically upward, both equations can be integrated quite easily in 1-d.

All the parameters like height, time etc are shown in the program. From this, after solving the kv or kv^2 case analytically, it will be possible to say which one matches your results when you put 90 deg in the program. Maybe the air resistance is not either of these in the program, but some other power or function of v.

-----------------------------------------------------

Hi Dr.Brain,

Don't we all wish that things were just always so simple...
 
Last edited:

1. What is projectile motion?

Projectile motion is the motion of an object through the air or space under the influence of gravity. It follows a curved path known as a parabola.

2. How is air resistance related to projectile motion?

Air resistance, also known as drag, is a force that opposes the motion of an object through the air. In projectile motion, air resistance affects the speed and trajectory of the object, causing it to deviate from a purely parabolic path.

3. What is the equation for projectile motion with air resistance?

The equation for projectile motion with air resistance is:

Fdrag = -1/2 * ρ * v^2 * A * Cd

where Fdrag is the drag force, ρ is the density of the air, v is the velocity of the object, A is the cross-sectional area of the object, and Cd is the drag coefficient.

4. How does air resistance affect the range of a projectile?

Air resistance reduces the range of a projectile by slowing down its horizontal velocity. This is because the drag force acts in the opposite direction of the motion, causing the projectile to lose speed and fall to the ground sooner.

5. What are some factors that can affect the amount of air resistance on a projectile?

The amount of air resistance on a projectile can be affected by factors such as the shape and size of the object, the velocity of the object, the density of the air, and the presence of any turbulence or wind. Additionally, the drag coefficient can vary depending on the surface properties of the object and the fluid it is moving through.

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