# Homework Help: Projectile Motion problem involving air resistance

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1. Jan 25, 2017

1. The problem statement, all variables and given/known data
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:
$$\\ m=0.7\text{kg} \\ k=0.01 \frac{\text{kg}}{\text{m}} \\ \theta=30 \degree$$

2. Relevant equations
$$F_{air}=kv^2$$

3. The attempt at a solution
I divided the dynamics of the problem into x-component and y-component equations:
From $$F_{x}=-kv_x^2$$, I got:
$$x\left(t\right)=\frac{m}{k}\ln \left(kv_it-m\sec \theta \right)$$

I divided the y-component of the motion into two parts--going up and going down:
I used $$F_{y1}=-mg-kv_y^2$$ for the going up part, and I used $$F_{y2}=-mg+kv_y^2$$
solving the differential equation for the going up part, I got:
$$v_y\left(t\right)=\sqrt{\frac{mg}{k}}\tan \left(\arctan \left(v_{yi}\sqrt{\frac{k}{mg}}\right)-t\sqrt{\frac{g}{m}}\right)$$ -- 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 $$v_y(t)$$ of the going down part and integrating it, I got:
$$y\left(t\right)=-\frac{m}{k}\ln \left(\cosh \left(t\sqrt{\frac{gk}{m}}\right)\right)+y_i$$

Can someone help me solve this problem?

2. Jan 25, 2017

### ehild

No, this is not correct. The force of air resistance is opposite to the velocity vector and its magnitude is proportional to v2. v2 is a scalar, it does not have components.

3. Jan 26, 2017

### Ray Vickson

You have not given an initial speed, so your problem is not completely specified.

For any given initial speed you can set up and solve the DEs numerically, and I think that is about the best you can do (in view of the criticism of "ehild" in #2).

Perhaps, though, you can get usable approximations by attempting something like a perturbation theory approach in which you essentially expand in powers of the small parameter $k$.

4. Jan 26, 2017

Sorry, I forgot to mention that-- the initial velocity is 9 m/s

5. Jan 26, 2017

What do you mean? Can't any force in vectorspace be divided into
[PLAIN]https://wikimedia.org/api/rest_v1/media/math/render/svg/18d95a7845e4e16ffb7e18ab37a208d0ab18e0e0, [Broken] https://wikimedia.org/api/rest_v1/media/math/render/svg/3dc8de3d8ea01304329ef9518fad7a6d196c4c01 components?

Last edited by a moderator: May 8, 2017
6. Jan 26, 2017

### Ray Vickson

The friction force acts along the negative of the tangent to the trajectory. See, eg.,
http://wps.aw.com/wps/media/objects/877/898586/topics/topic01.pdf
or
http://young.physics.ucsc.edu/115/range.pdf

The case of $\vec{f}_{\text{friction}} = - k \vec{v}$ is tractable, but not your case of $\vec{f}_{\text{friction}} = - k |v|^2\, \vec{v}/|v| = -k |v| \vec{v}$.

Last edited by a moderator: May 8, 2017
7. Jan 28, 2017