# Drag in the x and y directions

• Hermes Chirino
In summary, the conversation discusses two examples in a book about Classical Mechanics where the velocity and position of a particle moving in the x-direction and y-direction are solved using differential equations. The drag force in the medium is proportional to the velocity. The y-direction has an extra force of gravity, while the x-direction does not. The question is raised about why the premise for the x-direction does not include an extra force, similar to the premise for the y-direction. The reason given is that the mass is used for mathematical purposes and is not relied upon heavily. It is also noted that in a vacuum, gravity is the only force acting on a falling object. However, in air, the drag force affects the falling objects, causing them to fall
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 ?

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

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

Nathanael said:
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}##
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 really 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?

Hermes Chirino said:
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?

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?

## 1. What is drag in the x and y directions?

Drag in the x and y directions, also known as air resistance, is a force that opposes the motion of an object in the horizontal and vertical direction caused by the friction between the object and the air molecules it moves through.

## 2. How does drag affect the motion of an object?

Drag in the x and y directions can slow down the motion of an object because it acts in the opposite direction of the object's movement. It can also cause the object to change direction or trajectory.

## 3. What factors affect the amount of drag in the x and y directions?

The amount of drag in the x and y directions is affected by the shape and size of the object, the speed at which it is moving, and the density and viscosity of the air through which it is moving.

## 4. How can we reduce drag in the x and y directions?

To reduce drag in the x and y directions, we can change the shape of the object to make it more aerodynamic, decrease its speed, or change the properties of the air it is moving through, such as reducing its density or viscosity.

## 5. How is drag calculated in the x and y directions?

The drag force in the x and y directions can be calculated using the drag equation, which takes into account the object's size, speed, and air properties. It can also be measured experimentally using wind tunnels or other testing methods.

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