What is the Maximum Altitude of a Model Rocket?

[Januz Johansen]

Homework Statement

Hello all, I have to determine the maximum height of a model rocket.
The rocket is launched at an 45 degree angle
The finale velocity is 15,69 m/s (v)
the extutevelocety is 14,23 m/s (c)
The total mass is 180 gram (m) or m0)
The emty mass is 52 grams (mf)
The mass pr sek is 0,45 kg/s (me)
And the total burntime is 0,28 sek (t)

Homework Equations

How can i calculate the height at burnout, (i know how to calculate the last part) or all in one?

The Attempt at a Solution

I have tryed intregrating the rocket equation (v(t)) to get s(t), so i can calulate it with phytagoras, i got the equation from v(t) to:

but i have looked around to see it it was right, and i found this equation:

they give me different results, who is right, or are we both wrong?

Hope you can help

 

[TJGilb]

What did you use as your initial differential equation for the acceleration of the rocket $(\frac {d^2x} {dt^2})$?

 

[Januz Johansen]

What did you use as your initial differential equation for the acceleration of the rocket ($\frac {d^2x} {dt^2}$)? Or did you start with velocity ($\frac {dx} {dt}$)?

Hello, Yes sorry forgot, i used the velocity

so dx/dt, and its the same for the example i found

 

[TJGilb]

So you know velocity is the derivative of position with respect to time, but in your solution it looks like you integrated with respect to mass (or maybe turned time into a function of mass), am I correct?

 

[Januz Johansen]

So you know velocity is the derivative of position with respect to time, but in your solution it looks like you integrated with respect to mass (or maybe turned time into a function of mass), am I correct?

I see in the examble i found he rewrite the equation to

and do this

but will this make the difference, it should be the same (m(t) =mf at the burnout time, because there are no more fuel right?)

 

[TJGilb]

When deriving the rocket equation, you start with the sum of forces to get $\sum F =thrust-mg=ma$ which can be rewritten as $\frac {d^2x} {dt^2} = \frac {thrust-mg} m$. But since mass is a function of time in rockets, it can be replaced by $\frac {d^2x} {dt^2} = \frac {thrust-{m(t)}g} {m(t)}$. For the record thrust is the mass flow rate multiplied by the exhaust velocity.

It actually looks like they simplify it a little by not including the initial mass in their equation for m(t). But if you use the correct mass equation of $m(t)=m_0-mfr*t$ then solve for time you'd get $t= \frac {m(t)-m_0} {mfr}$ where the mfr is the mass flow rate. If you replaced every value of t with that "stuff" and integrated with respect to it, that could work. But that would be really overcomplicating the integration.

 

[TJGilb]

Or going off what they use for m(t) you could integrate with to $\frac {m(t)} {-m_e}$ after plugging that into every t value.

 

[TJGilb]

I was actually looking at what you got a bit closer, and it might be good. I'd recommend replacing every t in their equation with your function for m(t), simplifying, and see if it matches what you got.

 

[Januz Johansen]

I was actually looking at what you got a bit closer, and it might be good. I'd recommend replacing every t in their equation with your function for m(t), simplifying, and see if it matches what you got.

Ok thank you. I will try that, and make an update post (if you have a little time later) :D

 

[Januz Johansen]

I was actually looking at what you got a bit closer, and it might be good. I'd recommend replacing every t in their equation with your function for m(t), simplifying, and see if it matches what you got.

Sorry i said m(t)=-me*t but it's m(t)=m0-me*t
My mistake