Gravitation – IIT/JEE Solved Problems
Phase 1 • Page 5 • Mixed Numericals & Final Recap
Problem 21: Height Where Orbital Speed is Half
Q: At what height above Earth’s surface does the orbital speed of a satellite become half of its value near the surface?
Solution:
Orbital speed v = √(GM/r)
vh / v0 = √(R / (R + h)) = 1/2
Squaring:
R / (R + h) = 1/4
R + h = 4R
h = 3R
Key Method: Always compare ratios before substituting values.
Problem 22: Change in Time Period with Radius
Q: If radius of orbit of a satellite becomes four times, how does its time period change?
Solution:
Time period T ∝ r3/2
T2 / T1 = (4)3/2 = 8
New time period = 8 times original
IIT Favourite: This proportionality is tested repeatedly.
Problem 23: Satellite Speed vs Energy (Concept)
Q: A satellite in a lower orbit has less total energy than in a higher orbit. Yet, it moves faster. Explain.
Answer:
Total energy of satellite is negative. Lower orbit means more negative energy. However, orbital speed increases as radius decreases (v ∝ 1/√r). Hence, lower orbit → higher speed but lower total energy.
Exam Insight: Speed and energy do not vary in the same way.
Problem 24: Satellite Falling Due to Energy Loss
Q: A satellite loses a small amount of energy due to atmospheric drag. What happens to its orbit and speed?
Answer:
Loss of energy causes satellite to move to a lower orbit. Orbital radius decreases and speed increases. Eventually, it may burn in atmosphere.
Problem 25: Escape Velocity vs Orbital Velocity
Q: Find the ratio of escape velocity to orbital velocity of a satellite at the same radius.
Solution:
v = √(GM/r)
ve = √(2GM/r)
ve / v = √2
One-Line Result: Escape velocity is √2 times orbital velocity.
PHASE-1 MASTER THINKING METHOD
- Always convert height/depth → distance from centre
- Use proportionality first, numbers last
- Remember three key relations:
- v ∝ 1/√r
- T ∝ r3/2
- E ∝ −1/r
- Check physical meaning of answer
PHASE-1 FINAL TAKEAWAY
If you can relate radius, speed, time period, and energy, you can solve most Gravitation problems without heavy calculation.
Phase 1 • Gravitation Completed Successfully ✅
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- Gravitation – Part 1
- Gravitation – Part 2
- Gravitation – Part 3
- Gravitation – Part 4
- Gravitation – Part 5
- Gravitation – Part 6
- Gravitation – Part 7
- Gravitation – Part 8
- Gravitation – Part 9
- Gravitation – Part 10
- Gravitation – Part 11
- Gravitation – Part 12
- Gravitation – Part 13
- Gravitation – Part 14
- Gravitation – Part 15
- Gravitation – Part 16
- Gravitation – Part 17
- Gravitation – Part 18
- Gravitation – Part 19
- Gravitation – Part 20
- Gravitation – Part 21
- Gravitation – Part 22
- Gravitation – Part 23
- Gravitation – Part 24
- Gravitation – Part 25
- Gravitation – Part 26
- Gravitation – Part 27
- Gravitation – Part 28
- Gravitation – Part 29
- Gravitation – Part 30
- Gravitation – Part 31
- Gravitation – Part 32
- Gravitation – Part 34
- Gravitation – Part 35
- Gravitation – Part 36
- Gravitation – Part 37
- Gravitation – Part 38
- Gravitation – Part 39
- Gravitation – Part 40
- Gravitation – Part 41
- Gravitation – Part 42
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