Q.Match Column A item 'Parsec' with the correct item in Column B.
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🔒 Start your 14-day free trial to unlock the full solution →Concept understanding — Relative Motion Parallax
Relative Motion Parallax: Seeing Depth Through Motion
Close one eye and hold your thumb up at arm's length. Now move your head from side to side. Notice how your thumb appears to jump back and forth against the distant wall, while the wall itself barely seems to move. That's relative motion parallax in action — and you've been using it your whole life without realising.
The Intuition: Why Things "Slide" Past Each Other
Imagine you're sitting in a moving train, looking out the window. The trees near the track whip past in a blur. The mountains far away barely seem to shift. The moon? It appears to follow you.
This isn't an illusion — it's geometry. When you move, objects at different distances sweep across your field of view at different angular speeds. The closer an object is, the faster its image moves across your retina (or camera sensor). The farther it is, the slower it moves.
This is why you can judge distance even with one eye closed — as long as you move your head. Parallax works with one eye; stereopsis (3D vision) needs two.
The Precise Statement
Relative motion parallax is the apparent shift in the relative position of two objects when the observer changes their viewpoint. The amount of shift is inversely proportional to the distance of the objects from the observer.
Mathematically, for an object at distance d from an observer moving sideways by a distance b (the baseline), the angular displacement θ (in radians) is approximately:
θ≈db
This is for small angles. The key insight: if you have two objects at distances d1 and d2, the difference in their angular shifts tells you which is closer and by how much.
Relative angular shift=b(d11−d21)
The sign of this difference tells you direction: the nearer object appears to move opposite to your motion relative to the farther one.
Why It Matters for Exams
You'll encounter relative motion parallax in three contexts:
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Astronomy: Measuring distances to nearby stars. Earth's orbit provides the baseline b (2 AU). A star's apparent shift against distant background stars gives its distance via d=b/θ (with θ in radians).
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Robotics and computer vision: A moving camera can reconstruct depth by tracking how image features shift between frames. …
Why this formula?
Relative Motion Parallax: Why the Formula Holds
Relative motion parallax is a powerful depth cue: nearby objects appear to move faster across your field of view than distant ones when you (or your viewpoint) move sideways. Let's build the formula from first principles.
1. The Physical Setup
Imagine you are moving with velocity v perpendicular to your line of sight. You fix your gaze on a distant reference point (say, a mountain far away). Now consider an object at distance d from you.
- Key idea: As you move, the direction to the object changes. This angular shift per unit time is the parallax rate.
2. Deriving the Angular Speed
Let the object be at perpendicular distance d from your path. When you move a small distance Δx sideways, the object's angular position θ (measured from your forward direction) changes.
From simple geometry:
tanθ=dx
where x is your lateral displacement from the point of closest approach.
Differentiate with respect to time:
sec2θ⋅dtdθ=d1⋅dtdx
Since dtdx=v (your speed) and sec2θ=1+tan2θ=1+(dx)2, we get:
dtdθ=dv⋅1+(x/d)21
3. The Key Formula (for small angles)
When the object is nearly straight ahead (x≈0), the formula simplifies dramatically:
dtdθ=dv
This is the core result: the angular speed of a stationary object due to your motion is inversely proportional to its distance.
4. Why This Makes Intuitive Sense
- Near objects (d small): dv is large → rapid angular motion.
- Far objects (d large): dv is small → almost no apparent motion.
This is exactly what you see from a moving car: telegraph poles whiz by, but mountains barely move.
5. The Parallax Shift Over a Finite Displacement
If you move a total distance L sideways, the total angular shift (parallax angle) is:
Δθ=dL(for small angles, in radians)
This is the parallax formula used in astronomy (where L is the Earth's orbital diameter and d is the star's distance).
6. Why the Formula Holds — The Core Reasoning
| Step | What happens | Why it matters |
|------|--------------|----------------| …
A parsec is a very large astronomical unit of distance, used to measure distances to stars. …
Parsec is a unit of distance (1 parsec ≈ 3.26 light years), used in astronomy for stellar distances.
The name 'parsec' itself comes from 'parallax of one arc-second' — it is the distance at which 1 AU subtends an angle of one arcsecond. It is purely a u …
- CBSE 2022Set ANNUAL1 markMCQQ.Match Column A item 'Parsec' with the correct item in Column B.(a) kg m^2(b) Poise(c) Cp - Cv = R(d) E = mc^2(e) 1/frequency(f) distance(g) 24 hours
›Reveal solutionSolution
Parsec is a unit of distance (1 parsec ≈ 3.26 light years), used in astronomy for stellar distances.
The name 'parsec' itself comes from 'parallax of one arc-second' — it is the distance at which 1 AU subtends an angle of one arcsecond. It is purely a u …
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