Projectile Motion Using Differential Equations

Hi everyone. It seems I've forgotten how to solve this one. I was wondering if someone could help me remember how.

Think projectile motion, straight up/down no resistance. I point out the part I have a problem with near the end. (this is all in accordance with a book on applied mathematics.)

**d**^{2}x/dt^{2}= -g initial conditions are **dx/dt=v**_{0} x(0) = 0

**x(t) = -1/2*g*t**^{2} + v_{0} *t

The book makes says that I can solve for t to get

**t=v**_{0}/g - how?

and then realize the next equation without any explanation.

**X**_{max}= v_{0}^{2}/2g - how?

I would be very grateful for help and direction on this one. Thanks.

Re: Projectile Motion Using Differential Equations

Since we are given:

$\displaystyle x(t)=-\frac{1}{2}gt^2+v_0t$

then differentiating with respect to $\displaystyle t$, we find:

$\displaystyle v(t)=-gt+v_0$

Equating this to zero (at the apex of the trajectory which is $\displaystyle x_{\text{max}}$, the velocity will be zero), we find:

$\displaystyle -gt+v_0=0$

$\displaystyle t=\frac{v_0}{g}$

Now, using this value for $\displaystyle t$, we find:

$\displaystyle x_{\text{max}}=x\left(\frac{v_0}{g} \right)=-\frac{1}{2}g\left(\frac{v_0}{g} \right)^2+v_0\left(\frac{v_0}{g} \right)=$

$\displaystyle -\frac{v_0^2}{2g}+\frac{v_0^2}{g}=\frac{v_0^2}{2g}$

Re: Projectile Motion Using Differential Equations

"A pattern so grand and complex..."

Can you explain v(t) from x(t)

Thanks Mark.... BTW I'm a HUGE Rush fan.

Re: Projectile Motion Using Differential Equations

By definition, we have:

$\displaystyle v(t)\equiv\frac{d}{dt}x(t)$

This simply means that velocity is the time rate of change of displacement, or position.

So, if you have the displacement, you may differentiate it to get the velocity.

I was saddened a bit the other day to hear "Fly By Night" used in a commercial to sell Volkswagens. (Worried)(Surprised)

Re: Projectile Motion Using Differential Equations

LOL... to all of your last post... I must have missed that because it is late/early here... Can't believe I didn't see Calc I.

Yeah, that commercial is a bit... off.

Thank you again Mark. I can get back to studying in the morning. Good to know that there are great people on this sight with great answers.

Re: Projectile Motion Using Differential Equations

By the way, I was raised in Evansville...I really miss the Fall Festival there!

Re: Projectile Motion Using Differential Equations

The colors are wonderful this year.

Re: Projectile Motion Using Differential Equations

BTW, how are you generating the equation graphics? They look great.

Re: Projectile Motion Using Differential Equations

Those are generated using $\displaystyle \LaTeX$. Do a search here and online for LaTeX usage, and you will soon be making nice looking expressions too!