# Drag in the x and y directions

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1. Jun 29, 2015

### Hermes Chirino

I'm reading this book on Classical Mechanics and there are two examples in the book where we are asked to find one expression for the velocity $v$, and one for the position $x$, both as functions of time for a particle moving in the x-direction in a "medium" where the drag force is proportional to $v$. We are also asked to find velocity and position in the y-direction, same medium, drag force propotional to $v$. They use differential equations methods to solve it. I don't have any trouble understanding their methods or how they got the equations; my question is on their premises.

For the x-direction, they use as premise:
$$ma_x=m\frac{dv}{dt}=-kmv_x$$

I understand the drag force $-kmv$ is equal in magnitude to the force $ma$, but opposite direction.

For the y-direction, they use as premise:
$$F_T=m\frac{dv}{dt}=-mg-kmv_y$$

I understand the total force is equal to the force of gravity $mg$, minus the drag force pointing in the opposite direction.

My question here is: Why they don't use a similar premise for the x-direction, something like:
$$F_T=ma_x-kmv_x$$

What is the diference ? What I am missing here ?

2. Jun 29, 2015

### Nathanael

Shouldn't it be $ma_x=m\frac{dv_x}{dt}=-kv_x$ ?
Otherwise you're implying that deceleration due to drag would be independent of mass (but clearly feathers fall slower than heavy things).

That is not similar to what they did in the y-direction. $F_T$ is, by definition, equal to $ma_x$ right? So what sense does this equation make?

In the y-direction they equated $ma_y$ with $F_{net.y}$ and in the x-direction they equated $ma_x$ with $F_{net.x}$.
The y-direction just happens to have an extra force (gravity) that contributes to $F_{net.y}$

3. Jun 29, 2015

### Hermes Chirino

They use m, so when solving for the differential equation, the math comes a little bit easier, they mention in the book that we shouln't realy too much on the mass, they use it only for mathematical purposes. I understand your point in the y direction, that gravity comes as an extra force, but when an object is in free fall, isn't gravity the only force?

4. Jun 29, 2015

### SteamKing

Staff Emeritus
Only when things are falling in a vacuum.

Why do you think that a cannon ball and a feather dropped from the same height in air hit the ground at different times?