Perturbation with equations of motion for air resistance

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SUMMARY

The discussion focuses on applying perturbation methods to analyze the motion of a ball tossed upwards with an initial speed V_0, considering both gravity and air resistance modeled as -mkv^2. The participants clarify that the first step involves solving the equation of motion without air resistance, yielding v(t) = -gt + v_0. The introduction of the air resistance term leads to the equation a = -g - kv^2, where the perturbation parameter is identified as kv_0^2/g. The conversation highlights the importance of correctly interpreting the forces acting on the ball, particularly the opposing nature of gravity and air resistance.

PREREQUISITES
  • Understanding of Newton's second law (F=ma)
  • Familiarity with differential equations
  • Knowledge of perturbation methods in physics
  • Basic concepts of air resistance and its mathematical representation
NEXT STEPS
  • Study the perturbation method in classical mechanics
  • Learn about solving differential equations with air resistance
  • Explore the concept of perturbation parameters in physics
  • Investigate the effects of varying air resistance on projectile motion
USEFUL FOR

Students and educators in physics, particularly those studying classical mechanics, as well as researchers interested in the mathematical modeling of motion under the influence of forces like gravity and air resistance.

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Homework Statement


"A ball is tossed upwards with speed [tex]V_0[/tex]. Air resistance is [tex]-mkv^2[/tex] and there's gravity too.

Find the the time it takes the ball to reach the maximum height. Do not solve the equation of motion exactly. Use the perturbation method on the equation of motion. Solve the equation of motion without the air resistance. Then include the air resistance term, but plug in the first solution to the air resistance term. Find the leading order correction to x.

Homework Equations


F=ma

The Attempt at a Solution


It's unclear what is meant by 'solve' the E.O.M. I assume that meant find v(t).
Without air resistance, that was quick: [tex]v = -gt + v_o[/tex]. Then I added the air resistance term (and canceled mass):
[tex]a = -g -kv^2[/tex]
Plugging in the above into that for v didn't seem to suggest anything -- in fact after integrating I got an upward parabola which doesn't seem to make sense.

I think the perturbation parameter might be [tex]\frac{kv_o^2}{g}[/tex] since that's unitless.
 
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Never heard of 'perturbation parameter'. But the diff. eq. is easy to solve.

Your equation for a is wrong. g and v^2 oppose each other (unless you considered g < 0)
 

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