(a) A rocket is fired straight up from a tower 110 ft above the ground with an initial velocity of 224 ft/s. (Assume the acceleration due to gravity is −32 ft/s2).
When does the rocket reach its maximum height? and at this time, how far is the rocket above the ground?

Question

(a) A rocket is fired straight up from a tower 110 ft above the ground with an initial velocity of 224 ft/s. (Assume the acceleration due to gravity is −32 ft/s2).
When does the rocket reach its maximum height? and at this time, how far is the rocket above the ground?

bibby · Answer

@zepp
@AravindG
@tcarroll010

richyw · Answer

integrate the acceleration with respect to time to get the velocity, the constant of integration is the initial velocity. then integrate this velocity function with respect to time, the constant of integration will be the initial height. 

then you have the position as a function of time. 

to figure out the maximum height you just need to solve for when v(t)=0, then plug that t value into the position function to get the height!

bibby · Answer

also one more thing. When it asks for "find displacement when acceleration is 0", dpo I also plug back into s(t)?

richyw · Answer

uh, they told you to assume that acceleration is -32...

richyw · Answer

maybe that's after it hit the ground?

bibby · Answer

A different problem.

zepp · Answer

Okay, this is a physics question, but whatever.

Given the initial height, let's call it $\large y_0$, 110 ft; We are looking for the final height, $\large  y$. We also know that when the rocket will reach its highest point, it's velocity is 0, $\large v=0$, and the acceleration would be $\large -32ft/s^2$.

From the free-fall equations of kinematics, using the formula:
$\large v^2=v_0^2+2a(y-y_0)$

So, 
$\large 0=(244ft/s)^2-2a(y-110ft)\
\large 59536=2a(y-110)\\large
930.25=y-110\\large y=930.25+110=1040.25$

The final height would be 1040.25 feet in the sky.

zepp · Answer

If you want to find $t$, then plug everything in $\large y=y_0+\frac{1}{2}(v_0+v)t$ and solve for $t$.

bibby · Answer

zepp: that's the thing. It's one of my previous calc finals and I've never seen it worded like this although I'm pretty sure integration is how to go about it.

zepp · Answer

Of course, it would be solved using integration, but that's a classical mechanics introductory course question ;x

bibby · Answer

I guess I meant the technique and wording and not solving. I'm sure derivatives aren't exclusive to calculus :D.

richyw · Answer

$$a=\frac{dv}{dt}$$$$\int a\,dt=\int dv$$$$v=at+v_0$$$$v=\frac{dx}{dt}$$$$\int (at+v_0)dt=\int dx$$$$x=\frac{1}{2}at^2+v_0t+x_0$$

zepp · Answer

^ could easily be derived using simple algebra :D

richyw · Answer

show me?

zepp · Answer

Let's go from the acceleration, which is $$\large a=\frac{\Delta v}{\Delta t}=\frac{v_f-v_i}{t_f-t_i}$$If we set $t_i=0$ and $t_f=t$, we get $\large a=\frac{v-v_0}{t}$, or $\large v=v_0+at$

Now the formula to find the displacement is $\large \Delta x=x-x_0$

zepp · Answer

YES FINALLY IT LOADED

zepp · Answer

|dw:1355785810205:dw|

We know that the displacement could be given by $\large \Delta x = v_{av}\Delta t=\frac{(v_f - v_i)}{2}$
So the distance would be $\large x= x_0 + \frac{1}{2}(v_0+v)t$

Now, if you substitute $v=v_0+at$ into the formula above, you'll get $\large x = x_0+v_0t+\frac{1}{2}at^2$ by algebraic manipulations :)

bibby · Answer

ah sorry I closed the question. I don't think pm will work out but good luck if you guys can still type

richyw · Answer

you can type after it's closed anyways! and yeah but this works because acceleration is constant right? personally i'd just do it the faster way! (i can never memorize the kinematic eqns though)

zepp · Answer

Yes, acceleration is constant, it's the gravitational pull, usually, so :P