# Particle falling in earth's gravity with air resistance.

• kraigandrews
In summary, a particle falls in Earth's gravity and experiences a resistive force of F = -αv^2. To determine the constant α, one can solve the differential equation or use the net force balance at terminal speed.
kraigandrews

## Homework Statement

A particle falls in Earth's gravity. The force of air resistance is F = −αv v where v is the velocity, v is the speed, and α is a constant. (The direction of the force is opposite to the velocity, and the magnitude of the force is αv2.) If the terminal speed is 20.2 m/s and the mass is 2.6 kg, determine the constant α.

F=ma=mdv/dt

## The Attempt at a Solution

I was hoping somebody could point me in the right direction. I know how to solve a problem with just a resistive force, however, I am not sure where to start when given both the speed and velocity vector in the retarding force. So just any guidance as to where to start would be greatly appreciated.
Thanks.

You could solve this by solving the differential equation that way you'll get the x(t) function, and from there you can calculate the constant

or, you could think about the force balance after some time when particle reaches it's final speed

Start with F = ma. What are the expressions for the individual forces that act on the particle? What is the net force F that acts on the particle?

Ok, so how would I incorporate the vector v and the speed v into the diff eq?

Last edited by a moderator:
you can divide it into components, and then solve the differential equation for each component, that's harder way than just writing the net force balance for the time when speed reaches it's final speed, but you'll learn more from it;)

## 1. How does air resistance affect the speed of a falling particle?

Air resistance, also known as drag, is a force that opposes the motion of an object through air. As a falling particle moves through air, it experiences a force of air resistance that increases as its speed increases. This means that air resistance slows down the particle's speed, causing it to fall at a slower rate.

## 2. What factors influence the amount of air resistance on a falling particle?

The amount of air resistance on a falling particle is influenced by several factors, including the size and shape of the particle, the density and viscosity of the air, and the speed and direction of the particle's motion. Generally, larger and more aerodynamic objects experience less air resistance than smaller and less streamlined objects.

## 3. How does air resistance affect the acceleration of a falling particle?

As a falling particle experiences air resistance, its acceleration decreases. This is because air resistance is a force that opposes the particle's motion, and according to Newton's second law of motion, the net force on an object is equal to its mass multiplied by its acceleration. So, as air resistance increases, the net force on the particle decreases, and therefore its acceleration decreases as well.

## 4. Can air resistance ever cause a falling particle to stop moving?

No, air resistance cannot cause a falling particle to completely stop moving. This is because as the particle falls, it gains kinetic energy, which is the energy of motion. As the particle moves faster, its kinetic energy increases, and this energy is constantly being used to overcome the force of air resistance. So while air resistance may slow down the particle's speed, it cannot bring it to a complete stop.

## 5. How can the effects of air resistance on a falling particle be calculated?

The effects of air resistance on a falling particle can be calculated using equations that take into account the factors mentioned earlier, such as the particle's size and shape, air density and viscosity, and its speed and direction of motion. These equations allow scientists to predict the position, velocity, and acceleration of a falling particle at any given time, taking into account the effects of air resistance.

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