Special relativity in projectile motion

Question

I'm trying to describe projectile motion wrt a reference frame $S'$ that moves with velocity $v=0.6c$ in the $x$ direction wrt to the reference frame $S$ where the projectile was launched. I know projectile motion equations and Lorentz transformations but I'm kind of struggling to understand what is it that $S'$ sees. In $S$ I know $$x(t) = x_0 + v_o\cos(\alpha)t$$ $$y(t) = y_0 + v_o\sin(\alpha)t-\frac{g}{2}t^2$$ And Lorentz transformations are $$ \left[ {\begin{array}{c} x' \\ y' \\ t' \end{array} } \right] = \underbrace{\left[ {\begin{array}{ccc} \gamma & 0 &-v\gamma\\ 0 & 1 & 0 \\ -v\gamma/c^2 & 0 & \gamma \\ \end{array} } \right]}_{L_{xv}} \left[ {\begin{array}{c} x(t) \\ y(t) \\ t \end{array} } \right]$$ Now here are my questions,

• Is the last equation correct? (Correct in the sense that I simply have to substitute the expressions for $x(t)$ and $y(t)$) If yes, then $x',y'$ would depend on $t$ and shouldn't they depend on $t'$ instead?
• I am quite intrigued by the angle $\alpha$. Shouldn't I also apply some sort of transformation $\alpha\rightarrow\alpha'$? because $S'$ won't see the same angle that $S$ does, right?
• Also should I make a velocity transformation for the initial velocity and somehow account for it in $S'$? If that's the case how would I need to approach and add this to the equations of $S'$?
• Lastly, if I were to calculate the range and maximum height that $S'$ sees (which in $S$ are $R = x_0+\frac{v_0^2}{g}\sin(2\alpha)$ and $h_{max} = y_0 + v_0^2\frac{\sin(\alpha)^2}{2g}$ respectively) since $y' = y$ then $h_{max}':$ maximum height seen from $S'$, should be the same right? And for $R':$ range seen from $S'$, would I need to find $t'$ (or $t$ not even sure) such that $y' = 0$ and substitute this value of $t'$ (or $t$) in $x'$ to get $R'$? Or would it be possible I use some transformation for $\alpha$, $v_0$ and $x_0$ and get $R'$ that way?

Any information or help would be greatly appreciated.

Answer 1

Yes, your matrix equation is correct, although we normally put the time component at the top, not the bottom.

You don't actually need to worry about transforming $g, v_o, \alpha$. Once you have equations for $x', t'$ in terms of $x, t$, you can substitute in $$x = x_0 + v_o\cos(\alpha)t$$ and then eliminate $t$, which will let you write $x'$ as a function of $t'$.

For small $v_o$, the projectile's motion in the $S'$ frame is almost a straight line going backwards at 0.6c. If $v_o$ is large, then your basic projectile motion equations for $x, y$ in $S$ are probably inadequate, unless you happen to have a gigantic plane with uniform gravity. ;)

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I won't do the substitution I mentioned above, but here's a simpler example to give you the general idea. Instead of parabolic motion, the projectile has simple uniform motion: $$x = ut$$

with $y=0$, so we can ignore $y$ and $y'$.

The Lorentz transformation gives us $$x' = \gamma(x-vt)$$ $$ct' = \gamma(ct-vx/c)$$

We have $v=\frac35 c$, therefore $\gamma=\frac54$. So $$x'=\frac54 x - \frac34 ct$$ $$ct'=\frac54 ct - \frac34 x$$ Substituting $x = ut$, $$x'=\left(\frac54 u - \frac34 c\right)t$$ $$ct'=\left(\frac54 c - \frac34 u\right)t$$ Thus $$t = \frac{ct'}{\frac54 c - \frac34 u}$$ Substituting that into our last equation for $x'$ we get $$x'=\left(\frac{\frac54 u - \frac34 c}{\frac54 c - \frac34 u}\right)ct'$$ or $$x'=\left(\frac{5u - 3c}{5c - 3u}\right)ct'$$

We can write that as $$x'=u't'$$ where $$u'=\left(\frac{5u - 3c}{5c - 3u}\right)c$$ Note that for small $u$, $u'\approx u-v$

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The Lorentz transformation is symmetrical, and it can often be useful to work with the inverse form,

$$x = \gamma(x'+vt')$$ $$ct = \gamma(ct'+vx'/c)$$

Here's a more general way to find $u'$.

$$x' = \gamma(x-vt)$$ $$t' = \gamma(t-vx/c^2)$$

Now $u'=x'/t'$, thus $$u'=\frac{x-vt}{t-vx/c^2}$$

Substituting $x=ut$ $$u'=\frac{ut-vt}{t-uvt/c^2}$$

$$u'=\frac{u-v}{1-uv/c^2}$$

which can also be written in this form: $$\frac{u'}c=\frac{\frac uc-\frac vc}{1-\frac uc \frac vc}$$

And due to symmetry, $$u=\frac{u'+v}{1+uv/c^2}$$

This is the well-known law of addition of relativistic velocities.