Fundamentals of Orbits and Launch Vehicles

Bank exams love testing the same skill in two different costumes — and the SBI PO Number Series section has a favourite trick. Once you have figured out the rule of one series, the very next question hands you a second series that starts with a new number but follows the exact same rule. The candidate who hunts for a brand-new pattern wastes 90 seconds. The candidate who recognises the format finishes both questions in under a minute.

Definition: The Second-Question Pattern is an SBI PO number-series format where a complete sample series is given (so its rule can be cracked), followed by an incomplete series that begins with a different first term but obeys the same arithmetic rule. You are then asked for a specific position (typically the 3rd, 4th or 5th term).

Why This Format Exists

The examiner is not testing whether you can crack two patterns. They are testing whether you can transfer a rule from one starting point to another and whether you can correctly count steps. Both are bread-and-butter banking skills — applying the same EMI formula to different loan amounts, applying the same depreciation rule to different machines. The number-series wrapper is just dressing.

The Core Method

The technique is brutally simple in three moves:

Step 1 — Read the sample series end to end and extract the rule. Is each term the previous one + n, x n, ^2, or a chain like x2 + 1, x3 + 2, x4 + 3? Confirm the rule by checking it across at least two consecutive jumps in the sample.

Step 2 — Take the new first term from the question series. Do not borrow the original first term. This is where most students lose marks: muscle memory pulls them back to the sample's starting number.

Step 3 — March the rule forward the required number of steps. Compute term by term, writing each on the rough sheet so you do not lose count. Stop at the asked position.

Memory aid: "Learn the rule, change the seed." The plant grows the same way, but it sprouts from a different seed.

Counting Steps Correctly

This is the silent killer of the format. "Find the 3rd term" does NOT mean apply the rule 3 times. The first term is given. Reaching the 2nd term takes 1 application. Reaching the 3rd term takes 2 applications. In general:

To reach the nth term from the first term, apply the rule (n - 1) times.

A small table makes this unforgettable.

Why it matters: One miscounted step turns a correct rule into a wrong answer, and the wrong answer is always sitting there as a distractor in the option set. The examiner deliberately places the off-by-one number among the four options.

Worked Example

Question:
Sample series: 3, 7, 15, 31, 63, 127
New series first term: 5. Find the 3rd term.

Solution:
Step 1: Find the rule from the sample. 3 -> 7 (x2 + 1), 7 -> 15 (x2 + 1), 15 -> 31 (x2 + 1). Confirmed: each term = previous x 2 + 1.
Step 2: Take the new first term = 5. Apply the rule (3 - 1) = 2 times to reach the 3rd term.
Step 3: 1st application — 5 x 2 + 1 = 11. So the 2nd term = 11.
Step 4: 2nd application — 11 x 2 + 1 = 23. So the 3rd term = 23.
Conclusion: The 3rd term of the new series is 23.

Real-world example: Imagine a fixed-deposit scheme that doubles your money each year and adds Rs 1 as a loyalty bonus. If Aman starts with Rs 3 and Bhavna starts with Rs 5, after the same number of years their growth rule is identical, but their balances are different because the seed amount is different. The bank question is literally this scenario in disguise.

Common misconception: "The new series must follow a new rule because the first term is different." Wrong. The whole point of the format is that the rule is shared. A different starting number does not change the multiplication, addition or chain pattern operating on it. The setter is testing rule-transfer, not rule-discovery.

Speed Tips for the Exam Hall

Because you only need 2 to 4 steps, manual computation beats calculator memory. Write the first term, draw an arrow, write the rule above the arrow, write the next term. Repeat. Even a chain rule like "x1 + 1, x2 + 2, x3 + 3, x4 + 4" can be marched out in under 25 seconds with practice.

If the rule of the sample is something complicated like differences forming a new series (e.g., differences are 4, 8, 16, 32 — themselves doubling), the same nested rule applies to the new series. The first difference is now applied to the new starting number, then the next difference, and so on. The seed changes; the engine does not.

Orbital velocity (to stay in circular orbit near Earth): v = √(GM/r) ≈ 7.9 km/s for LEO. Escape velocity (to break free of Earth's gravity): v = √(2GM/R) ≈ 11.2 km/s at Earth's surface. Key relation: escape velocity = √2 × orbital velocity. Escape velocity is independent of the object's mass and direction; depends on the planet's mass and radius. Memory aid: '11.2 to escape, 7.9 to orbit.' First cosmic velocity = orbital (7.9), second cosmic velocity = escape (11.2), third cosmic velocity ≈ 16.7 km/s (to escape the solar system from Earth). For the Moon escape velocity is only ~2.4 km/s due to low mass.

PSLV (Polar Satellite Launch Vehicle): 4-stage, alternating solid-liquid stages; workhorse for SSO/polar and small GEO payloads; launched Chandrayaan-1 and Mangalyaan. GSLV Mk II: 3-stage with indigenous Cryogenic Upper Stage (CUS); ~2.5-tonne GTO payload. LVM3 (earlier GSLV Mk III): heaviest, 3-stage (2 solid boosters S200 + liquid L110 core + cryogenic C25); ~4-tonne GTO / 8-tonne LEO; launched Chandrayaan-2, Chandrayaan-3, and crewed Gaganyaan. SSLV (Small Satellite Launch Vehicle): for small satellites up to ~500 kg to LEO, 'launch-on-demand.' Memory aid: cryogenic engines use liquid hydrogen (fuel) + liquid oxygen (oxidiser), giving high specific impulse for heavy GTO missions.

Cover the answer, recall, then check. 12 cards on must-know facts for UPSC Prelims.

Q1. What is the escape velocity from Earth's surface?
A1. About 11.2 km/s — the minimum speed to break free of Earth's gravity without further propulsion.

Q2. At what altitude is the geostationary orbit, and what is its period?
A2. ~35,786 km above the equator; orbital period 24 hours; inclination 0° — the satellite appears fixed over one point.

Q3. Difference between geostationary and geosynchronous orbit?
A3. Both have a 24-hour period, but geostationary is a special case at 0° inclination over the equator (appears stationary); a geosynchronous orbit can be inclined.

Q4. What is a Sun-synchronous orbit (SSO)?
A4. A near-polar low-Earth orbit in which the satellite passes over a given location at the same local solar time each day — ideal for remote sensing.

Q5. Which orbit is used for communication satellites and why?
A5. Geostationary orbit (~35,786 km) — the satellite stays fixed relative to the ground, so dish antennas need not track it.

Q6. What is a Geostationary Transfer Orbit (GTO)?
A6. A highly elliptical intermediate orbit used to move a satellite from low-Earth orbit up to the geostationary orbit.

Q7. What is India's workhorse launch vehicle?
A7. The PSLV (Polar Satellite Launch Vehicle) — reliable for polar/Sun-synchronous and small payloads; famously launched 104 satellites in one flight (2017).

Q8. What is a cryogenic engine?
A8. A rocket engine using propellants liquefied at very low temperatures — liquid hydrogen (fuel, ~ -253°C) and liquid oxygen (oxidiser, ~ -183°C); high efficiency but hard to master.

Q9. Which Indian vehicles use a cryogenic upper stage?
A9. GSLV and LVM3 (GSLV Mk III). PSLV does not have a cryogenic stage.

Q10. What is India's heaviest launch vehicle and its payload?
A10. LVM3 (GSLV Mk III) — ~4,000 kg to GTO and ~8,000 kg to low-Earth orbit; used for Chandrayaan-2/3 and Gaganyaan.

Q11. What are Lagrange points?
A11. Five points in a two-body system where gravitational and centripetal forces balance, so a small object can stay put. L1 hosts Aditya-L1; L2 hosts JWST.

Q12. What is the approximate orbital velocity of a satellite in low-Earth orbit?
A12. About 7.9 km/s (the "first cosmic velocity") — needed to maintain a circular LEO.

Worked Example

Problem/Question: Consider the following statements about satellite orbits:

1. A geostationary satellite orbits at about 36,000 km above the equator with a period of 24 hours.
2. Low Earth Orbit (LEO) satellites are preferred for remote sensing and Earth observation.
3. A geostationary orbit is a type of polar orbit.
   Which of the statements given above are correct?
   (a) 1 and 2 only (b) 2 and 3 only (c) 1 and 3 only (d) 1, 2 and 3

Solution/Model answer:

• Statement 1: A geostationary satellite sits at **35,786 km (36,000 km)** over the equator, matching Earth's ~24-hour rotation. Correct.
• Statement 2: LEO (a few hundred km) gives high resolution and is preferred for remote sensing/Earth observation. Correct.
• Statement 3: A geostationary orbit is equatorial (0° inclination), not polar. Statement 3 is wrong.
  Eliminate options with 3: (b), (c), (d) drop. Only (a) remains.

Answer/Takeaway: (a) 1 and 2 only.