# Projectile motion with air resistance

How do we write differential equations for projectile motion in 2 dimensions featuring air resistance of magnitude kv^2, acting directly opposite to the direction of motion at that moment in time, where v is the velocity in the direction of motion at that moment in time?

## Answers and Replies

tiny-tim
Homework Helper

Tell us what you think, and why, and then we'll comment!

Obviously I'm asking because I don't know. More pertinently I can't imagine how to model this properly. Presumably we'll need independence on the x and y axes, connected with time.

For free-fall I could model it pretty easily:

$$mg - kv^2 = m \cdot \frac{dv}{dt}$$

But this doesn't seem to be remotely of the same difficulty. Velocity in free fall is always in the same direction as acceleration, but in the projectile motion case, the velocity is defined by the initial projection angle and velocity (which would be given of course, along with the values of m, g and k).

Nugatory
Mentor
but in the projectile motion case, the velocity is defined by the initial projection angle and velocity (which would be given of course, along with the values of m, g and k).

Try writing the differential equation in terms of vectors $\vec{v}$ and $\vec{F}=m\vec{a}$. That will at least get the problem modeled properly, and you can decompose it into coupled differential equations for ##x(t)## and ##y(t)##.

The initial angle and speed provide the boundary conditions you'll need to determine the arbitrary constants that show up in the solutions of the differential equations.

(And I have to caution you that solving these equations is a non-trivial problem).

Last edited:
Try writing the differential equation in terms of vectors $\vec{v}$ and $\vec{F}=m\vec{a}$. That will at least get the problem modeled properly, and you can decompose it into coupled differential equations for ##x(t)## and ##y(t)##.

Well ok, perhaps this is an analogous equation to mine for free-fall, where v has now been replaced by a vector?

Question is, how to decompose this into my x(t) and y(t) differential equations?

jhae2.718
Gold Member
Question is, how to decompose this into my x(t) and y(t) differential equations?

Equate the components of the acceleration and force vectors to obtain the scalar equations of motion.

tiny-tim
Homework Helper

(just got up :zzz:)
For free-fall I could model it pretty easily:

$$mg - kv^2 = m \cdot \frac{dv}{dt}$$

ok, same, but with vectors …

$-mg\mathbf{y} - kf(\mathbf{v})\mathbf{v} = m \cdot \frac{d\mathbf{v}}{dt}$
where ##f(\mathbf{v})## = … ?

Equate the components of the acceleration and force vectors

What do you mean?

ok, same, but with vectors …

$-mg\mathbf{y} - kf(\mathbf{v})\mathbf{v} = m \cdot \frac{d\mathbf{v}}{dt}$
where ##f(\mathbf{v})## = … ?

I'm not sure I understand what that y is doing there. As for f(v), not sure ... looks like it should just be v to me, but I'm not sure why you wrote it as such then ...

arildno
Homework Helper
Gold Member
Dearly Missed
f is a scalar, non-negative function.
What ought f to be then?

tiny-tim
Homework Helper
I'm not sure I understand what that y is doing there.

because gravity is mg directly downwards,

so the vector for the force of gravity is -mg in the y direction, ie -mg time the unit vector in the y direction, ie -mgy
As for f(v), not sure ... looks like it should just be v to me

yes, but you need to write it in terms of the vector v, so it's -(k√(v2))v

Redbelly98
Staff Emeritus
Homework Helper
Equate the components of the acceleration and force vectors to obtain the scalar equations of motion.

What do you mean?
jhae means Fx = m ax , and similarly for the y-component. That's two equations of motion, one for each component.

As for the original question:

The air resistance force has a magnitude k v2, and direction opposite to that of v. For the x-component of the force, multiply this magnitude by the cosine of the angle it makes w.r.t. the +x-direction -- this is -vx/v -- and this gives you the x-component of the force due to air-resistance.

Do the same for the y-component.

And then you'll have the force components due to air resistance to use in the equations relating Fx to ax and Fy to ay.

p.s. This is worth repeating:
(And I have to caution you that solving these equations is a non-trivial problem).

Ok, I don't know how to include the angles - surely with 2 equations we can only afford to have 2 variables, x and y?

So far I have

$$F_x = m \cdot \frac{d^2x}{dt^2} = -k \cdot \frac{dx}{dt}$$

and

$$F_y = m \cdot \frac{d^2x}{dt^2} = -mg \cdot cos(\theta) - k \cdot \frac{dx}{dt}$$

I think there is a problem in how I have resolved the weight though. How can I do this better?

Redbelly98
Staff Emeritus
Homework Helper
You're getting there . Yes, there are just the two variables you mentioned to concern yourself with.

A couple of problems to clear up in your equations:

1. The weight is -mg. There is no cosine term involved in the weight, since it always acts downward, in the -y-direction, regardless of the angle of the trajectory.

2. Also, you have incorrectly multiplied kv2 and -vx/v -- so you are missing a factor of v in the air resistance expression. (And the same error occurs in the y-equation).

arildno
Homework Helper
Gold Member
Dearly Missed
You won't be able to solve this analytically, but it is a good exercise to set up the equations of motion that will govern the system.

1. The weight is -mg. There is no cosine term involved in the weight, since it always acts downward, in the -y-direction, regardless of the angle of the trajectory.

Ok then so maybe

$$F_y = m \cdot \frac{d^2y}{dt^2} = -mg - k \cdot \frac{dy}{dt}$$

2. Also, you have incorrectly multiplied kv2 and -vx/v -- so you are missing a factor of v in the air resistance expression. (And the same error occurs in the y-equation).

I'm not sure I understand ... by v, do you mean the resultant of vx and vy, and by vx you mean dx/dt? If so then maybe:

$$F_y = m \cdot \frac{d^2y}{dt^2} = -mg - k \cdot \frac{dy}{dt} \cdot ((\frac{dy}{dt})^2+(\frac{dx}{dt})^2)^{1/2}$$

and

$$F_x = m \cdot \frac{d^2x}{dt^2} = - k \cdot \frac{dy}{dt} \cdot ((\frac{dy}{dt})^2+(\frac{dx}{dt})^2)^{1/2}$$

arildno
Homework Helper
Gold Member
Dearly Missed
In your last last line, the "dy/dt" outside the root should be replaced with "dx/dt"

Redbelly98
Staff Emeritus
Homework Helper
I'm not sure I understand ... by v, do you mean the resultant of vx and vy, and by vx you mean dx/dt?
Yes to both.

Apart from arildno's correction, you got it.

Thank you.

And the boundary conditions would involve me specifying initial y and x displacement from the origin as well as initial x velocity and y velocity (which can be found as the cos and sin components respectively of the initial total magnitude of velocity, making sure the y initial velocity is positive if the point is travelling upwards initially and negative if it is travelling downwards initially), and nothing else?

Nugatory
Mentor
Thank you.

And the boundary conditions would involve me specifying initial y and x displacement from the origin as well as initial x velocity and y velocity (which can be found as the cos and sin components respectively of the initial total magnitude of velocity, making sure the y initial velocity is positive if the point is travelling upwards initially and negative if it is travelling downwards initially), and nothing else?

Yes.