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## Homework Statement

An 8mm wide cone of a hypothetical substance which does not melt, with a mass of 2.7 grams, is shot straight up at a 90 degree angle. Only accounting for fluid drag and gravity, what are the initial velocities required for it to:

a) reach a height of 55000 m

b) be travelling at 50% of its initial velocity at that height

c) travel there in 1 second

## Homework Equations

As far as I know:

[itex]F_D=1/2\rho v^2C_DA[/itex]

[itex]F=MA[/itex]

## The Attempt at a Solution

My attempt was to create an equation for displacement vs time:

[itex]\delta =\rho C_DA/2M[/itex]

[itex]v=v_i-(\delta v^2+g)t[/itex]

After isolating v:

[itex]v=\sqrt{\frac{v_i}{\delta t}-g\delta+(\frac{1}{2\delta t})^2}-\frac{1}{2 \delta t}[/itex]

After integrating it, I got (ignore the vector stuff, I don't know why I put that there):

Which is far as I can go without a mathematica to isolate d for me.

Is there a specific method to go about this, to make it simpler? Or do I need to use a mathematica (don't have one atm, not really sure how to use them)? Or is this all completely wrong?