(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that, in the limit of large damping, the time of flight of a projectile (the projectile is fired from level ground) is approximately,

[tex]t \approx \left(\frac{w}{g} + \frac{1}{\gamma}\right)\left(1-e^{-1-\gamma w/g}\right)[/tex]

2. Relevant equations

The equation of motion for the projectile is given by,

[tex]m\ddot{r} = -\gamma mr -mg[/tex]

so that,

[tex]\ddot{z} = -\gamma \dot{z} - g[/tex]

The solution to this equation is,

[tex]z = \left(\frac{w}{g} + \frac{g}{{\gamma}^2}\right)\left(1-e^{-\gamma t}\right) - \frac{gt}{\gamma}[/tex]

3. The attempt at a solution

I arrived at the answer by saying that the,

[tex]e^{-\gamma t}[/tex]

term vanishes for large gamma, so setting z = 0 i have,

[tex] t = \frac{w}{g} + \frac{1}{\gamma}[/tex]

which you can then substitute back in to the exponetial term that vanished for t --- this gives the answer but the method seems terrible to me, i just saw it.

I thought of expanding the exponential about t = 0 but i'm not getting there.

I think i'm missing something obvious. It's looking at making approximations to solutions is this question and i think i'd find it pretty useful once i actually get throught it, so i'd appreciate your help.

Thank you.

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# Homework Help: Projectile and time of travel

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