In a particular scene of a science fiction movie, a UFO is flying horizontally at a very high altitu

Q: Why do we ignore relativistic corrections like time dilation?

The problem explicitly states to ignore them to focus purely on the classical 'light travel time' effect, which is often the dominant factor in apparent superluminal motion.

Q: Why is the apparent speed different from the actual speed?

Because the UFO is moving, each successive light pulse has a different distance to travel to reach you. This changing delay compresses or stretches the time interval between pulses as you perceive them.

Q: Can the apparent speed be greater than the speed of light?

Yes! If the UFO is moving towards you at a speed close to $c$, the pulses are 'bunched up' so tightly that the apparent speed can exceed $c$. This is an optical illusion, not a violation of relativity.

Q: How did we get $\sin\theta = \frac{x}{r}$?

By looking at the right triangle formed by the observer, the point directly below the UFO, and the UFO itself. The angle $\theta$ is with the vertical, so the opposite side is the horizontal distance $x$.

Q: What does $\frac{dt'}{dt}$ physically represent?

It represents the ratio of the time interval between receiving two pulses to the time interval between emitting them. If it's greater than 1, you see the events happening in slow motion.

Question

In a particular scene of a science fiction movie, a UFO is flying horizontally at a very high altitude with a speed $u$ that is $\eta$ ($< 1$) times of speed $c$ of light. The UFO is emitting sharp light pulses at regular and very small intervals. Find the speed of the UFO recorded by an observer on the ground at point O when the UFO appears at an angle $	heta$ with the vertical. Ignore relativistic corrections.
An UFO (un-identified flying object) is believed to be a space ships used by aliens.

Sigma Percentile · Accepted Answer

### Analyzing the Setup
Imagine you are standing on the ground, looking up at the night sky. A UFO zips past horizontally at a staggering speed $u$, which is a significant fraction of the speed of light $c$ (specifically, $u = \eta c$).

The UFO is emitting light pulses at regular intervals. However, because the UFO is moving, each successive pulse has a different distance to travel to reach your eyes. This changing distance means the light pulses will take different amounts of time to reach you, creating an optical illusion regarding the UFO's speed.

### The Master Equation
Let's set up a coordinate system. You, the observer, are at the origin $(0,0)$. The UFO is flying at a constant altitude $H$. At any given time $t$, its horizontal position is $x(t)$.

The distance $r(t)$ from the UFO to you is given by the Pythagorean theorem:
$$r(t) = \sqrt{H^2 + x(t)^2}$$

When the UFO emits a pulse at time $t$, that light must travel the distance $r(t)$ at speed $c$. Therefore, you receive the pulse at a later time $t'$:
$$t' = t + \frac{r(t)}{c}$$

### The Time Dilation Effect
To find out how the time interval between pulses changes, we differentiate the reception time $t'$ with respect to the emission time $t$:
$$\frac{dt'}{dt} = 1 + \frac{1}{c} \frac{dr}{dt}$$

We need the rate of change of the distance, $\frac{dr}{dt}$. By differentiating our distance equation $r^2 = H^2 + x^2$, we get:
$$2r \frac{dr}{dt} = 2x \frac{dx}{dt}$$

Since the UFO's speed is $u = \frac{dx}{dt}$, this simplifies to:
$$\frac{dr}{dt} = \frac{x}{r} u$$

Looking at the geometry of the problem, the angle $	heta$ is between the vertical and the line of sight. In this right triangle, the opposite side is $x$ and the hypotenuse is $r$. Thus, $\sin	heta = \frac{x}{r}$. Substituting this back, we find:
$$\frac{dr}{dt} = u \sin	heta$$

Now, we plug this into our time derivative equation. Knowing that $u = \eta c$, we get:
$$\frac{dt'}{dt} = 1 + \frac{u \sin	heta}{c} = 1 + \eta \sin	heta$$

### Final Calculation
The apparent speed $v_{	ext{app}}$ you observe is the actual distance the UFO moves divided by the time interval you perceive. Using the chain rule:
$$v_{	ext{app}} = \frac{dx}{dt'} = \frac{\frac{dx}{dt}}{\frac{dt'}{dt}}$$

Substitute our known values:
$$v_{	ext{app}} = \frac{u}{1 + \eta \sin	heta}$$

**This is our final answer!**

Notice something fascinating: if the UFO were flying *towards* you, $x$ would be negative, and the apparent speed would be $\frac{u}{1 - \eta \sin	heta}$. If $\eta$ is close to 1, this apparent speed can actually exceed the speed of light! This mind-bending effect is known as **superluminal motion** and is frequently observed in the jets of distant quasars.

Visualized Solution

$C oor d ina t e S e t u p$

$D i s t an ce t o O b ser v er$

$T im eo f R ece pt i o n$

$A pp a r e n tT im e I n t er v a l$

$R a t eo f C han g eo f D i s t an ce$

$G eo m e t r i c S u b s t i t u t i o n$

$T im eD i l a t i o n F a c t or$

$A pp a r e n tS p ee d$

$T h e W a y F or w a r d$

The Sigma Insight: Relative Velocity

Analyzing the Setup Imagine you are standing on the ground, looking up at the night sky

The Master Equation Let's set up a coordinate system

The Time Dilation Effect

Final Calculation The apparent speed $v_{app}$ you observe is the actual distance the UFO moves divided by the time interval you perceive

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A boat which has a speed of $5 km/h$ in still water crosses a river of width $1 km$ along the shortest possible path in $15 min$ . The velocity of the river water in km/h is

Visualized Solution

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } RateofChangeofDistance

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } TimeDilationFactor

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ApparentSpeed

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } TheWayForward