# Newtonian Mechanics - Particle in Motion with Air Resistance

• derravaragh
In summary, the time required to move a distance d after starting from rest is t = arctan(v/√((g/k)sinθ))/√(kgsinθ)
derravaragh

## Homework Statement

A particle of mass m slides down an inclined plane under the influence of gravity. If the motion is resisted by a force f = kmv^2, show that the time required to move a distance d after starting from rest is

t = [arccosh(e^(kd))]/√(kgsin(θ)

where θ is the angle of inclination of the plane.

## Homework Equations

F = ma
F_g = mgsin(θ)
Resistance = -kmv^2
Motion of particle => ma = mgsin(θ) - kmv^2

## The Attempt at a Solution

My attempt was to set up the equation for the motion (ma = mgsin(θ) - kmv^2) and use differential equations to solve. After dividing by mass, I had:
dv/dt = gsin(θ) - kv^2
which I then divided by k, and substituted (g/k)sinθ for C^2 giving

dv/kdt = C^2 - v^2

After collecting terms and integrating I came to

t = arctan(v/√((g/k)sinθ))/√(kgsinθ)

I thought I was on the right track as I have the numerator correct, but I do not know where to go from here, or if this is even correct so far. Any help would be much appreciated. Also, I know this question has been asked before, but the answer given didn't make sense to me so it doesn't help.

Try writing the acceleration as ##a = \frac{dv}{dt} = \frac{dv}{dx}\frac{dx}{dt} = v \frac{dv}{dx}## where ##x## is distance along plane. Separate variables and integrate to find ##v## as a function of ##x##. Then, writing ##v = \frac{dx}{dt}## you can separate variables again and integrate to find ##t## as a function of ##x##.

[EDIT: Actually, your way works too. However, I think there is a typo in your final expression for t. Did you mean to write arctanh rather than arctan? If you solve your (corrected) expression for ##v## and then let ##v = \frac{dx}{dt}## you can separate variables and integrate to find ##t## as a function of ##x##]

Last edited:
Yes, it was supposed to be arctanh (and also denominator was correct not numerator). I just finished this, thank you very much for your assistance.

## 1. What is Newtonian Mechanics?

Newtonian Mechanics is a branch of classical mechanics that describes the motion of particles under the influence of external forces.

## 2. What is a particle in motion with air resistance?

A particle in motion with air resistance refers to an object that is moving through a fluid, such as air, and is experiencing a force due to the resistance of the fluid. This phenomenon is also known as drag.

## 3. How is air resistance calculated?

Air resistance is calculated using the drag equation, which takes into account the density of the fluid, the velocity of the object, the cross-sectional area of the object, and the drag coefficient.

## 4. How does air resistance affect the motion of a particle?

Air resistance acts in the opposite direction of the object's motion, slowing it down and changing its trajectory. As the object's velocity increases, the force of air resistance also increases, eventually reaching a point where it is equal to the force of gravity, resulting in a constant velocity known as terminal velocity.

## 5. What are some real-life examples of Newtonian Mechanics - Particle in Motion with Air Resistance?

Some real-life examples of this concept include a skydiver falling through the air, a baseball being thrown, and a plane flying through the sky. In all of these cases, air resistance plays a role in the motion of the object.

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