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I am trying to understand projectile motion, but I am stuck. I'm trying to follow http://farside.ph.utexas.edu/teaching/336k/Newtonhtml/node29.html#e5.23u. However, I don't understand how they reach equation 178 by integrating equation 176. And similarly how they reach equation 181 by integrating equation 177.

EDIT: I've realised vt is a constant. Therefore you have integral of (1/vx)dt between vx0, v which = (-g/vt) * integral (1/dt).

So ln(v/vx0) = (-g/vt)*t + c.

So where does the c go?

F J
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  • Also, please use Mathjax:

    https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference

     – Alexander Geldhof Oct 26 '19 at 11:31
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    @AlexanderGeldhof Thanks, but why are they integrating between vx0 and v but then in the result they've used ln(vx) - ln(vx0) = ln(vx/vx0)? – F J Oct 26 '19 at 11:35
  • @AlexanderGeldhof I guess that was a mistake? But what actually does vx mean? – F J Oct 26 '19 at 11:37
  • "So where does the $c$ go?": Following your steps, note that the $c$ you have is actually $0$, because if $t=0$ then $v_x=v_{x0}$ so then you have $$\ln(1)=(-g/v_t)\cdot 0+c \implies c=0$$ As a result, we obtain the same result as equation 179. – projectilemotion Oct 26 '19 at 11:55

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Starting from equation 176:

$$ \frac{dv_x}{dt} = - g \frac{v_x}{v_t},$$

we move around the factors

$$ \frac{1}{v_x} \frac{dv_x}{dt } = - \frac{g}{v_t},$$

and we integrate both sides to $dt$, with boundaries $t = 0$ and $t = T$.

$$ \int_0^T \frac{1}{v_x} dt \frac{dv_x}{dt} = - g \int_0^T \frac{dt}{v_t}.$$

We can use the chain rule to change the integration variable in the left integral. This also changes the boundaries:

$$ \int_{v_x \mathrm{\ when\ } t = 0}^{v_x \mathrm{\ when\ } t = T} \frac{d v_x}{v_x} = - g \frac{(T - 0)}{v_t}.$$

Calculating the integral leaves us with

$$ \ln(v_x(T)) - \ln(v_x(0)) = -g \frac{T}{v_t}.$$

The $c$ is hidden in the $v_t$ (note its definition).

Also remark that your source has the horrible habit of denoting both variables and constants by the same letter ($v_x, t$), probably being the source for most of your confusion.

Alexander Geldhof
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