How Does Air Resistance Affect Projectile Motion?

[pimpalicous]

Homework Statement

A particle is projected vertically upward with a speed v₀ under the influence of a drag force that is proportional to the particle's speed. As the particle falls back down, its terminal speed is v_t.
(a) Find the time it takes to reach ymax

(b) Find ymax

(c)Show that your answers for a and b make sense by taking the limit where the drag force approaches zero. Do they reduce to the familiar eqns of 1st semester physics?

(d) evaluate a and b numerically for blah blah blah

Homework Equations

F_d=-k*m*v

The Attempt at a Solution

I got solutions for a and b pretty easily.

For a I wrote:
-mg-kmv=m[tex]\frac{dv}{dt}[/tex]
I separated and integrated

-[tex]\int dt[/tex]=[tex]\int dv/(g+kv)[/tex]
then
-t=[tex]\frac{1}{k}[/tex]*ln[[tex]\frac{g+kv}{g+kv_{0}}[/tex]]

I got a speed of
v=-[tex]\frac{-g}{k}[/tex]+([tex]\frac{g}{k}[/tex]+v₀)e^-kt

I set this equal to zero and solved for t to get

[tex]\Large t[/tex]=[tex]\frac{1}{k}[/tex]*ln[[tex]\frac{g+v_{0}k}{g}[/tex]]

For part b, I separated the v equation and integrated to find y

y(t)=-[tex]\frac{gt}{k}[/tex]+([tex]\frac{g}{k^{2}}[/tex] +[tex]\frac{v_{0}}{k}[/tex])[-e^(-kt) +1]

I plugged in the equation for t and got

ymax=-[tex]\frac{g}{k^2}[/tex]*ln[1+[tex]\frac{v_{0}k}{g}[/tex]]+([tex]\frac{g}{k^2}[/tex] + [tex]\frac{v_{0}}{k}[/tex])([tex]\frac{-g}{g+v_{0}k}[/tex] +1)

I checked them in c by taking the limit of t and ymax as k approached zero.

The t checked perfectly. I got t=[tex]\frac{v_{0}}{g}[/tex].

The ymax didn't go so well.

ymax=-[tex]\frac{g}{k^2}[/tex]*ln[1+[tex]\frac{v_{0}k}{g}[/tex]]+([tex]\frac{g}{k^2}[/tex] + [tex]\frac{v_{0}}{k}[/tex])(-g/(g+0) +1)

I used the approximation ln(1+x)=x-x^2/2 where v0k/g=x

ymax=-[tex]\frac{g}{k^2}[/tex][[tex]\frac{v_{0}k}{g}[/tex]-([tex]\frac{v_{0}^2*k^2}{2g^2}[/tex]]

it reduced to

ymax=-[tex]\frac{v_{0}}{k}[/tex]+[tex]\frac{v_{0}^2}{2g}[/tex]

The second term is what I'm looking for but the first term obviously goes to infinity and can't be right. Where did I make a mistake?

 

[Jhero]

Thank you for your post. It seems like you have made some good progress in solving this problem. Here are some suggestions and clarifications on your solution:

a) To find the time it takes to reach ymax, you can set y=ymax in your equation for y(t) and then solve for t. This will give you the same result as you have for t in your solution.

b) To find ymax, you can use the equation for y(t) and set t=0 (since the particle starts at the ground) to find the initial height. Then, you can set v=0 (terminal velocity) and solve for y to find ymax. This will give you the same result as you have for ymax in your solution.

c) In order to take the limit as k approaches zero, you need to use L'Hopital's rule. This will help you evaluate the limit of the natural logarithm as k approaches zero. Also, note that your expression for ymax should not have a negative sign in front, since the particle is moving upwards initially.

d) To evaluate a and b numerically, you can plug in values for m, g, k, and v0. Then, you can use a calculator or computer program to solve for t and ymax. It might also be helpful to plot the graphs of t vs. v0 and ymax vs. v0 to see how they vary with different initial velocities.

Overall, your approach to solving this problem is correct, but there are some small mistakes and clarifications needed. Keep up the good work and feel free to ask for further assistance if needed.