# Equations of motion in a free fall with friction

• torito_verdejo
In summary: Free fall means no external forces except gravity are acting, so that the only acceleration is ##g##. In the presence of friction, the object is not in free fall.

#### torito_verdejo

Homework Statement
A point mass ##m## is dropped from a hight ##z##. Its motion is subject to gravity force and a friction force ##F_f=-\lambda \dot{z}##.
Write the equations of motion for this system.
Relevant Equations
$$F_f=-\lambda \dot{z}$$
I'm stack at the very beginning. If I use Newton's second law to find acceleration and integrate until I find the position, I must face

$$v(t) = \int_0^t g-\frac{\lambda v}{m} dt'=gt-\frac{\lambda }{m} \int_0^t\frac{\partial z}{\partial t}dt$$

But this last term feels pretty weird. I don't know how to interpret it, even though

$$\frac{\partial z}{\partial t}dt = dz$$

If I extract velocity from friction, we get

$$v(t)=\frac{m}{\lambda}(g-a(t))$$

which once again defines a function by its derivative. Should I be solving a differential equation?

In any case, I face two possible ways and I wonder which one, if any, is the correct one. Thank you in advance for the help.

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The simple answer is that yes, this is about solving differential equations. Have you studied these?

Your last equation could better be written:

##\ddot{z} = g - \frac{\lambda}{m}\dot{z}##

Note: this is, by definition, the "equation of motion" for the system.

In the first part of your post you effectively integrated this, using the initial conditions, to get:

##\dot{z} = gt - \frac \lambda m z##

Note that your "weird" integral was just the integral of a derivative, which is given by the fundamental theorem of calculus. Note that the derivative should be an ordinary time derivative, not a partial derivative.

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torito_verdejo and Delta2
Yes you are called to solve a first order linear differential equation here. The first step is to gather the unknown function and all its derivatives to the same side of the equation and write the equation as $$\dot v+\frac{\lambda}{m}v=g$$.
Now we notice that if we multiply the equation by the factor ##e^{\frac{\lambda}{m}t}## then the LHS becomes the derivative of the product ##e^{\frac{\lambda}{m}t}v(t)## so we ll have
$$(e^{\frac{\lambda}{m}t}v(t))'=ge^{\frac{\lambda}{m}t}$$
and from this last equation you can integrate both sides and after some algebraic manipulation you can solve for the unknown function ##v(t)##.

torito_verdejo and PeroK
Apart from what has already been said, just a bit of terminology: If there is friction, the object is not in free fall.

torito_verdejo

## 1. What is an equation of motion in free fall with friction?

An equation of motion in free fall with friction is a mathematical representation of the motion of an object that is falling due to gravity and experiencing frictional forces. It takes into account the acceleration, velocity, and displacement of the object as it moves.

## 2. How is the equation of motion in free fall with friction different from the equation of motion in free fall without friction?

The equation of motion in free fall with friction includes an additional term for the force of friction, which opposes the motion of the object. This results in a decrease in the acceleration and velocity of the object compared to free fall without friction.

## 3. What factors affect the motion of an object in free fall with friction?

The motion of an object in free fall with friction can be affected by the mass of the object, the force of gravity, the coefficient of friction, and the surface area of contact between the object and the surface it is falling on.

## 4. How can the equation of motion in free fall with friction be used to predict the motion of an object?

By plugging in values for the initial conditions (such as initial velocity and displacement) and the relevant factors affecting the motion, the equation of motion in free fall with friction can be used to calculate the position, velocity, and acceleration of the object at any given time during its fall.

## 5. Can the equation of motion in free fall with friction be used for all types of objects and surfaces?

No, the equation of motion in free fall with friction is specifically designed for objects falling on a flat, horizontal surface with a constant coefficient of friction. It may not accurately predict the motion of objects falling on inclined surfaces or objects with significantly different shapes or surface areas.