- #1

Adrsya Rupam

- 4

- 0

## Homework Statement

Ok, so I am attempting to solve a projectile motion problem involving air resistance that requires me to find the total x-distance the projectile traverses before landing again.

Given:

[tex]

\\

m=0.7\text{kg}

\\

k=0.01 \frac{\text{kg}}{\text{m}}

\\

\theta=30 \degree

[/tex]

## Homework Equations

[tex]F_{air}=kv^2[/tex]

## The Attempt at a Solution

I divided the dynamics of the problem into x-component and y-component equations:

From [tex]F_{x}=-kv_x^2[/tex], I got:

[tex]x\left(t\right)=\frac{m}{k}\ln \left(kv_it-m\sec \theta \right)[/tex]

I divided the y-component of the motion into two parts--going up and going down:

I used [tex]F_{y1}=-mg-kv_y^2[/tex] for the going up part, and I used [tex]F_{y2}=-mg+kv_y^2[/tex]

solving the differential equation for the going up part, I got:

[tex]v_y\left(t\right)=\sqrt{\frac{mg}{k}}\tan \left(\arctan \left(v_{yi}\sqrt{\frac{k}{mg}}\right)-t\sqrt{\frac{g}{m}}\right)[/tex] -- which is where I am stuck on... because when I plugged in my given values, the graph doesn't look right as its t-intercept is greater than one which is found from solving this kinematically without air resistance.

solving the differential equation for [tex]v_y(t)[/tex] of the going down part and integrating it, I got:

[tex]y\left(t\right)=-\frac{m}{k}\ln \left(\cosh \left(t\sqrt{\frac{gk}{m}}\right)\right)+y_i[/tex]

Can someone help me solve this problem?