# Vertical Projectile with air friction

1. Nov 6, 2005

### madking153

hi,

If we throw a ball vertical upward, we can easily find the maximum height if ignore the friction..

So if we switch on the friction, how can we find the max height ?

2. Nov 6, 2005

### HungryChemist

It will be just like the way you would go find the maximum height the ball would reach without the friction. That is, you will start from F = ma but now instead of F being only mg, there will have to be another force term describing the friction. Most likely, it will be some constant times the velocity of the object. So, your equation of the motion will be

F = ma = -mg-cv

Now is a math problem.

3. Nov 6, 2005

### madking153

i had the same equation motion, but how we can find the max height ?

4. Nov 6, 2005

### Tide

You find the height by solving the differential equation.

5. Nov 6, 2005

### Integral

Staff Emeritus
This is the differential equation you need to solve.

$$m \ddot {x} = mg - c \dot{x}$$

6. Nov 6, 2005

### Dr Transport

I believe that the best way to solve this is to make the change of variables
$$\ddot{x} = \frac{dv}{dt} = \frac{dv}{dx}\frac{dx}{dt} = v\frac{dv}{dx}$$
and integrate from 0 to the height h then solve for h.

7. Nov 6, 2005

### krab

actually for a ball in air, the air resistance is turbulent and the equation will be:$$m \ddot {x} = -mg - c \dot{x}^2$$

8. Nov 6, 2005

### Integral

Staff Emeritus
I was thinking along those lines also but was not sure enough to make the correction.

9. Nov 6, 2005

### whozum

What satisfies the 'turbulent' condition?

10. Nov 6, 2005

### dx

If it is $$\dot{x}^2$$ , you would have to consider the horizontal velocity at each instant too.

11. Nov 7, 2005

### madking153

i will consider both conditions .. if the initial velocity is not too high - then is propotional to v , if not ( high velocity ) then is proportional to v^2...

i got a long a strange solution for max height...

12. Nov 7, 2005

### krab

It's turbulent if the Reynolds is greater than about 30. This happens at very low speeds, so you can safely ignore the laminar case.

13. Nov 9, 2005

### blackcode

Answer.

You can use intergral to solve this problem. At this time, the domail is from t1 to t2.

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