# Some weird particle problem I invented. Long read.

Hello, this is my first post, been lurking here for awhile.

I made up this physics problem which involves a particle in 2-D. The particle has an initial velocity (Vo) or initial kinetic energy (KEo) launched at some angle. Once the particle is fired the only forces acting on it are air friction, which is in the opposite direction to its velocity and the particles own weight.

Here's how I modeled the air friction:

The particles kinetic energy (KE) decreases according this function of time.

KE(t) = KEo e^(-c*t), where c is some constant (with inverse seconds for units, just to keep the units consitant), a percentage, between 1 and 0. I used the particles KE as behaving like a charged capacitor discharging through a resistor, that's how I thought of the decreasing exponential function. I don't know if this is an accurate way to model air friction, just doing this for fun.

Find: a function of the particles X-Y position according to time.

Do I sound crazy?

This is what I've done so far.

I found the particles speed as a function of time:

v(t) = sqrt(Vo^2*e^(c*t)), this is a magnitude

I then looked as the forces:

The weight: W = m*g, in the negative y direction, duh

I'm unable to find the air friction force. If I knew the magnitude of the air friction force with respect to time I'd be fine.

I then used F = dp/dt (differential of linear momentum with respect to time) on the particles velocity to be used to find how the direction of the particle is changed by the forces. I don't know if you can do this?

I found the particles linear momentum, m*v(t); than differentiated with respect to time to express it as a force. Can you do that, to find out how the direction of the particle is changed?

Alright enough babbling, anyone have any clue what I'm talking about?

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HallsofIvy
Homework Helper
No, you don't sound crazy (have you read OTHER posts in this forum?? Now some of those are CRAZY!) but you do sound confused.

Energy is a scalar quantity. Particals of the same mass moving at the same speed in different directions will have the same kinetic energy. There is no way to determine direction of motion just knowing the kinetic energy. There simply isn't enough information.

Originally posted by HallsofIvy
Energy is a scalar quantity. Particals of the same mass moving at the same speed in different directions will have the same kinetic energy. There is no way to determine direction of motion just knowing the kinetic energy.
Yes you're right.

However:

We can determine how the particles velocity is changed. Since, the only force that will change the particles direction is gravity, because the air friction can only change the particles speed since it's in the opposite direction to the particles velocity. The particles weight is appling a constant force straight down, which accelerates the particles in the negative y direction.

The particles potential engergy is changed. I forgot about potential energy. The particles PE is affecting the particles KE while the particles is traveling through the air. I think that's my hole. Which makes my function of KE meaningless.

I found a function of time and the angel alpha (the angle between the air friction force vector and the particles weight vector) that determines the particles acceleration in seperate x and y coordinates. But I'm sure it's wrong now.

If I change the problem, make it so the air friction takes KE out of the particle according to my function, I think it make things easier, or at least make sense. My definition of the problem didn't make sense. I'll keep working at it. It's all a learning expirence.

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enigma
Staff Emeritus
Gold Member
Welcome to the forums, frankR

First off, your model for air friction is not very accurate.

KE(t) = KEo e^(-c*t)
Would mean that as time -> infinity, the kinetic energy goes to zero. For an object in free-fall, KE never goes to zero. The KE will increase since the force of gravity is always greater than or equal to the drag force (unless the shape or profile changes). As the object falls, it will pick up speed until it hits terminal velocity. At that point, KE will be constant, but non-zero.

There is no closed form solution for the problem you are describing. You can't get a simple formula. You need to do it numerically.

x(t), y(t), Vx(t), Vy(t)

Then solve those in increments of dt.

x(n)=x(n-1)+Vx(n-1)*dt
y(n)=y(n-1)+Vy(n-1)*dt
Vx(n)=Vx(n-1)+ &Sigma; Fx(n-1) * dt
Vy(n)=Vy(n-1)+ &Sigma; Fy(n-1) * dt

The only trick now is to determine what the forces in X and Y are.

Drag force can be reasonably accurately modeled as -k*V. You will also have gravity in the -y direction.

Therefore,

Fx(n-1) = -k*Vx(n-1)
Fy(n-1) = -k*Vy(n-1) - mg

Easiest way to do something like this is on the computer. If you're interrested and know how to "speak" matlab, I can forward a file with an example of this along with an animation.

Thank you for that post Enigma, very informative. I've also come to the conclusion that the problem was impossible to solve without a numerical method and realized my model of air friction was very inaccurate; yet much was learned conceptually.

I used a similar technique as you've described in an experiment where I measured the mechanical equivalent of heat with a foot pump and a pressure cooker. A numerical method was needed to determine the work done by the foot pump. My fellow students thought I was crazy to use such technique but my results were very good, with only 4.85% error.

I’d like to take a look at that Mathlab file. Thanks again.