(a) A bright object is at position $(0, D)$ at time 0 , where $D$ is a very large positive number. The object moves toward the positive $x$-axis with constant speed $v<c$ at an angle $\theta$ from the vertical. Find parametric equations for the position of the object at time $t$.
(b) Let $s(t)$ be the distance from the object to the origin at time t. Then $L(t)=\frac{s(t)}{c}$ gives the amount of time it takes for light emitted by the object at time $t$ to reach the origin.
Show that $L^{\prime}(t)=\frac{1}{c} \frac{v^{2} t-D v \cos \theta}{s(t)}$.
(c) An observer stands at the origin and tracks the horizontal movement of the object. Light received at time $T$ was emitted by the object at time $t$, where $T=t+L(t)$. Similarly, light received at time $T+\Delta T$ was emitted at time $t+d t$, where typically $d t \neq \Delta T$. The apparent $x$-coordinate of the object at time $T$ is $x_{a}(T)=x(t)$. The apparent horizontal speed of the object at time $T$ as measured by the observer is $h(T)=\lim _{\Delta T \rightarrow 0} \frac{x_{a}(T+\Delta T)-x_{a}(T)}{\Delta T}$. Tracing back to time $t$, show that $h(t)=\lim _{d t \rightarrow 0} \frac{x(t+d t)-x(t)}{\Delta T}=$ $\frac{v \sin \theta}{T^{\prime}(t)}=\frac{v \sin \theta}{1+L^{\prime}(t)}$
(d) Show that $h(0)=\frac{c v \sin \theta}{c-v \cos \theta}$.
(e) Show that for a constant speed $v$, the maximum apparent horizontal speed $h(0)$ occurs when the object moves at an angle with $\cos \theta=\frac{U}{c}$. Find this speed in terms of $v$ and $\gamma=\frac{1}{\sqrt{1-v^{2} / c^{2}}}$
(f) Show that as $v$ approaches $c$, the apparent horizontal speed can exceed $c$, causing the observer to measure an object moving faster than the speed of light! As $v$ approaches $c$, show that the angle producing the maximum apparent horizontal speed decreases to 0 . Discuss why this is paradoxical.
(g) If $\frac{v}{c}>1$, show that $h(0)$ has no maximum.