This MCQ module is based on: Potential Energy Escape Velocity
TOPIC 31 OF 33
Potential Energy Escape Velocity
🎓 Class 11
Physics
CBSE
Theory
Ch 7 – Gravitation
⏱ ~14 min
🌐 Language: [gtranslate]
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Potential Energy Escape Velocity
7.8 Gravitational Potential Energy
In Chapter 6 we learned that potential energy near Earth's surface is \(U = mgh\) with reference at the ground. This works only when g is essentially constant. For motion through large altitudes or interplanetary flight, g varies with distance, so we need a more general expression.
Consider a unit mass moved from infinity (where we set U = 0 as our reference) to a distance r from a point mass M. The work done by the gravitational force when bringing the mass in from infinity to r is:
\[W = \int_\infty^r \vec{F}\cdot d\vec{r} = -\int_\infty^r \frac{GMm}{r^2}dr = -\frac{GMm}{r}\]
By definition, gravitational PE is the negative of this work:
Gravitational Potential Energy:
\[U(r) = -\frac{G M m}{r}\]
The negative sign signals that the gravitational force is attractive — work must be done against gravity to move m away from M, raising U toward zero at infinity.
Near Earth's surface (r ≈ R_E + h, with h ≪ R_E):
\[U \approx -\frac{GMm}{R_E}+mgh\]
The first term is a constant, so the change in PE is \(\Delta U = mgh\) — the familiar near-surface formula!
7.9 Gravitational Potential
The gravitational potential V at a point is the gravitational PE per unit test mass:
\[V(r) = \frac{U(r)}{m} = -\frac{GM}{r}\]
Like U, V is negative and tends to zero at infinity. The gravitational field is the negative gradient of V:
\[E_g = -\frac{dV}{dr} = -\frac{GM}{r^2}\;\hat{r}\]
If several masses are present, V is the algebraic sum of individual potentials (superposition for scalars).
| Quantity | Formula | Units | Type |
|---|---|---|---|
| Gravitational Force F | GMm/r² | N | Vector |
| Gravitational Field E_g | GM/r² | N/kg or m/s² | Vector |
| Gravitational PE U | −GMm/r | J | Scalar |
| Gravitational Potential V | −GM/r | J/kg | Scalar |
7.10 Escape Speed
If we project a stone vertically upward from Earth with low speed, it falls back. With higher speed it goes higher before returning. Is there a critical speed beyond which it never returns? Yes — the escape speed.
Using conservation of energy. Initial KE + Initial PE = Final KE + Final PE. For "just barely escape" we set final KE = 0 at infinity, and final PE = 0:
\[\frac{1}{2}mv_e^2 - \frac{GMm}{R_E} = 0\]
Escape velocity from a planet of mass M and radius R:
\[v_e = \sqrt{\frac{2GM}{R}}=\sqrt{2gR}\]
For Earth: \(v_e = \sqrt{2\times 9.8\times 6.4\times 10^6} = 11{,}200\,\text{m/s}\approx 11.2\,\text{km/s}\) (or about 40,300 km/h!).
| Body | Mass (kg) | Radius (m) | v_e (km/s) |
|---|---|---|---|
| Moon | 7.4×10²² | 1.74×10⁶ | 2.4 |
| Mars | 6.4×10²³ | 3.4×10⁶ | 5.0 |
| Earth | 6.0×10²⁴ | 6.4×10⁶ | 11.2 |
| Jupiter | 1.9×10²⁷ | 7.0×10⁷ | 59.5 |
| Sun | 2.0×10³⁰ | 7.0×10⁸ | 618 |
Interactive Simulation: Escape Speed Calculator
Adjust the planet's mass and radius (relative to Earth) and instantly see how the escape speed changes.
v_escape = 11.2 km/s
Set mass=320, radius=11.2 for Jupiter (~ 59.5 km/s)
Worked Example 1: Energy needed to escape Earth
A satellite of mass 200 kg is to escape Earth's gravity. Find the minimum kinetic energy required.
\[KE_{\min}=\frac{1}{2}m v_e^2 = \frac{1}{2}\times 200\times(11{,}200)^2\]
\[=\frac{1}{2}\times 200\times 1.25\times 10^8 \approx \boxed{1.25\times 10^{10}\,\text{J}}\]
That's 12.5 GJ — the energy released by ~300 kg of TNT.
Worked Example 2: Why the Moon has no atmosphere
Show that the rms speed of oxygen molecules at lunar surface temperature (300 K) is comparable to lunar escape velocity. Mass of O₂ = 5.3×10⁻²⁶ kg, k = 1.38×10⁻²³ J/K.
\[v_{rms}=\sqrt{\frac{3kT}{m}}=\sqrt{\frac{3\times 1.38\times 10^{-23}\times 300}{5.3\times 10^{-26}}}\approx 484\,\text{m/s}\]
That's only ~0.5 km/s, much less than v_e(Moon) = 2.4 km/s — so heavy O₂ would normally stay. However, lighter gases like H₂ (m = 3.3×10⁻²⁷ kg) reach v_rms ≈ 1.93 km/s, comparable to lunar escape. Over billions of years even slow molecules in the high-velocity tail of the Maxwell distribution escape, leaving the Moon airless.
Worked Example 3: Speed at infinity
A rocket leaves Earth's surface at 15 km/s. What will be its speed when very far away (at infinity)?
Energy conservation: \(\tfrac{1}{2}v_i^2 - GM/R = \tfrac{1}{2}v_\infty^2\)
\[v_\infty^2 = v_i^2 - v_e^2 = 15^2 - 11.2^2 = 225 - 125.4 = 99.6\,(\text{km/s})^2\]
\[v_\infty = \sqrt{99.6} \approx \boxed{9.98\,\text{km/s}}\]
The "excess" kinetic energy beyond escape stays as residual speed at infinity.
Activity 7.3 — Modelling a Gravity WellL3 Apply
Materials: Stretchy fabric (Lycra), large heavy ball (cricket/bowling), light marble, ring stand or four chairs.
- Stretch the fabric flat across the chairs/stand to form a horizontal sheet.
- Place the heavy ball at the centre — it creates a deep "well".
- Roll the marble across the sheet, both far and close to the ball; vary speed.
Predict: At what speed will the marble (a) orbit, (b) spiral in, (c) "escape" off the sheet?
Slow marbles spiral in (insufficient KE). At intermediate speeds the marble orbits in elliptical paths around the heavy ball. With enough speed it shoots off the sheet — this corresponds to exceeding escape velocity. The stretched fabric is a 2D analogy of Einstein's curved spacetime: mass warps space, and other objects follow the curvature.
Competency-Based Questions
ISRO scientists at Sriharikota plan a deep-space mission to escape Earth's gravity and travel to Mars. They need to optimise the launch energy and consider how escape velocity changes with planet properties.
Q1. Escape velocity from a planet of radius R and mass M is given by:L1 Remember
Answer: (b) √(2GM/R).
Q2. If a planet's radius is doubled keeping density constant, escape velocity will:L4 Analyse
M = (4/3)πR³ρ ∝ R³. So v_e = √(2GM/R) ∝ √(R³/R) = R. Doubling R doubles v_e. The relation v_e = R·√(8πGρ/3) is convenient when ρ is fixed.
Q3. Fill in the blank: Gravitational potential energy at infinity is taken as ________.L1 Remember
Zero. This is the universal reference point. All other values U(r) = −GMm/r are negative because bringing a mass from infinity releases energy.
Q4. True/False: If you double a satellite's launch speed (from any value), it always escapes. Justify.L5 Evaluate
FALSE. Escape depends on whether the new speed exceeds v_e. For example, doubling 3 km/s gives 6 km/s — still below Earth's 11.2 km/s. The required condition is v ≥ v_e, not "double the previous speed".
Q5. HOT: Design a strategy for a Mars mission that minimises the energy needed to reach Mars from Earth's surface, given that you must escape Earth, fly through space, and brake into Mars orbit.L6 Create
Sample strategy: (1) Use Earth's rotational speed by launching eastward from near the equator — adds ~0.46 km/s for free. (2) Time launch for a "Hohmann transfer" window when Earth and Mars are aligned — needs minimum extra v above v_e (~11.6 km/s). (3) During cruise, use gravitational slingshot from Venus or a coast to save fuel. (4) At Mars, use aerobraking (atmospheric drag) instead of all-rocket deceleration. This is how ISRO's Mangalyaan (2014) reached Mars with a remarkably small payload of fuel — making it the most cost-effective Mars mission in history (₹450 crore vs NASA's $671 million MAVEN).
Assertion–Reason Questions
(A) Both true, R explains A. (B) Both true, R does NOT explain A. (C) A true, R false. (D) A false, R true.
A: The gravitational PE of an Earth-satellite system is always negative.
R: Gravitational force is attractive and PE is measured relative to infinity, where U = 0.
(A). Both true; R correctly explains A.
A: Escape velocity depends on the mass of the projectile.
R: A heavier rocket needs more kinetic energy than a lighter one to escape.
(D). A is FALSE — v_e = √(2GM/R) is independent of projectile mass m (it cancels out). R is TRUE because KE = ½mv² does scale with m, but the speed required is the same for all masses.
A: The Moon has no atmosphere.
R: The lunar escape velocity is comparable to the thermal speed of gas molecules at lunar temperatures.
(A). Both true and R explains A. Over geological time, gas molecules in the high-speed Maxwell tail exceed v_e(Moon) = 2.4 km/s and escape, leaving the Moon airless.
Frequently Asked Questions - Potential Energy Escape Velocity
What is the main concept covered in Potential Energy Escape Velocity?
In NCERT Class 11 Physics Chapter 7 (Gravitation), "Potential Energy Escape Velocity" covers core principles and equations needed for board exam success. The MyAiSchool lesson explains the topic with definitions, derivations, worked examples, and interactive simulations. Key formulas and dimensional analysis are included to build conceptual depth and problem-solving skills aligned with the CBSE 2025-26 syllabus.
How is Potential Energy Escape Velocity useful in real-life applications?
Real-life applications of Potential Energy Escape Velocity from NCERT Class 11 Physics Chapter 7 include engineering design, satellite mechanics, sports biomechanics, transportation safety, and electrical/electronic devices. The MyAiSchool lesson links every concept to a tangible example so students see physics as a problem-solving framework for the physical world, not as abstract formulas.
What are the key formulas in Potential Energy Escape Velocity?
Key formulas in Potential Energy Escape Velocity (NCERT Class 11 Physics Chapter 7 Gravitation) are derived step-by-step in the MyAiSchool lesson. Students should memorize the final formula AND understand its derivation for full board marks. Each formula is listed with its dimensional formula, SI unit, applicability range, and common pitfalls. The Summary section at the end of each part includes a quick-reference formula card.
How does this part connect to other parts of Chapter 7?
NCERT Class 11 Physics Chapter 7 (Gravitation) is structured so each part builds on the previous one. Potential Energy Escape Velocity connects directly to neighbouring parts via shared definitions, units, and methodology. The MyAiSchool lesson cross-references related concepts with internal links so students can navigate the whole chapter as one connected story rather than disconnected fragments.
What types of CBSE board questions come from Potential Energy Escape Velocity?
CBSE board questions from Potential Energy Escape Velocity typically include: (1) 1-mark MCQs on definitions and formulas, (2) 2-mark short-answer derivations or applications, (3) 3-mark numerical problems with units, (4) 5-mark long-answer derivations followed by application. The MyAiSchool lesson tags each Competency-Based Question (CBQ) with Bloom level (L1-L6) so students know how to study for each weight.
How can students use the interactive simulation effectively?
The interactive simulation in the Potential Energy Escape Velocity lesson allows students to adjust input parameters (sliders or selectors) and see physical quantities update in real time. To use it effectively: (1) try extreme values to understand limiting cases, (2) compare with the analytical formula, (3) check unit consistency, (4) test special configurations from worked examples. The simulation reinforces conceptual intuition that pure formula manipulation cannot.
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