1. The Physical Concept

This problem deals with the time-of-flight delay of light signals. Because the distance between the source (UFO) and the observer changes while the light is traveling, the rate at which pulses arrive is different from the rate at which they were emitted.

This is effectively the kinematic basis for the Doppler effect. As the source moves away or changes angle, the distance each subsequent pulse must travel increases, “stretching” the observed time interval.

2. Observed Time Interval ($dt’$)

Let the UFO emit two pulses separated by a true time interval $dt$.

• Pulse 1: Emitted at $t$. Travels distance $r(t)$.
  Arrival time: $t’_1 = t + \frac{r(t)}{c}$
• Pulse 2: Emitted at $t+dt$. Travels distance $r(t+dt)$.
  Arrival time: $t’_2 = (t+dt) + \frac{r(t+dt)}{c}$

The observed interval is the difference in arrival times:

3. Rate of Change of Distance ($\frac{dr}{dt}$)

We need to relate the change in radial distance $r$ to the horizontal velocity $u$.

From geometry: $r^2 = x^2 + h^2$. Differentiating with respect to time:

Substituting $\frac{dx}{dt} = u$ and $\frac{x}{r} = \sin \theta$ (from the diagram):

Therefore, the time expansion factor is:

$$ dt’ = dt \left( 1 + \frac{u \sin \theta}{c} \right) $$

4. Apparent Speed

The observer calculates speed based on the distance the UFO actually moved ($dx = u \cdot dt$) divided by the observed time interval ($dt’$).

$$ u_{recorded} = \frac{dx}{dt’} = \frac{u \cdot dt}{dt’} $$ Substituting our expression for $dt’$: $$ u_{recorded} = \frac{u \cdot dt}{dt \left( 1 + \frac{u \sin \theta}{c} \right)} $$

Using $\eta = \frac{u}{c}$, we arrive at the final result: