Escape Velocity Examples: 47 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Escape Velocity.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Physics.

Concept Recap

Escape velocity is the minimum speed an object must have to escape a body's gravitational pull without further propulsion, ignoring air resistance.

It is the launch speed needed so gravity cannot pull the object back forever.

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Escape Velocity asks students to choose the object, list external interactions, and reason from the resulting force or torque pattern.

Common stuck point: Students often know a formula related to escape velocity but skip the recognition step: Have I isolated one system and listed the external forces or torques acting on it before applying a law? That leads to a correct-looking substitution attached to the wrong physical model.

Sense of Study hint: Ask: Have I isolated one system and listed the external forces or torques acting on it before applying a law?

Worked Examples

Example 1

medium

Compute the escape velocity from Earth using

G = 6.67\times 10^{-11}

M = 5.97\times 10^{24} \text{ kg}

r = 6.37\times 10^6 \text{ m}

Answer

v_{e} \approx 1.12 \times 10^{4} m/s

First step

GM = 6.67\times 10^{-11} \cdot 5.97\times 10^{24} \approx 3.98\times 10^{14}

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Example 2

medium

Saturn has

GM \approx 3.79\times 10^{16}

and

r \approx 5.82\times 10^7

m. Find its escape velocity.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Find the escape velocity from a body where

GM = 4\times10^{14}

and

r = 6.4\times10^6 \text{ m}

Example 2

easy

How does escape velocity compare to circular orbital speed at the same radius?

Example 3

easy

Does escape velocity depend on the mass of the escaping object?

Example 4

easy

Escape velocity means the object escapes without further propulsion. Does gravity become zero after escape?

Example 5

easy

Earth's escape velocity is about

11.2 \text{ km/s}

. A planet has the same radius but four times Earth's mass. Find its escape velocity.

Example 6

easy

A planet has the same mass as Earth but four times the radius. Find its escape velocity relative to Earth's

11.2 \text{ km/s}

Example 7

easy

Which formula gives escape velocity:

\sqrt{GM/r}

\sqrt{2GM/r}

Example 8

easy

The Moon's escape velocity is about

2.4 \text{ km/s}

. Could a

5 \text{ kg}

rock and a

50 \text{ kg}

rock both escape at that speed?

Example 9

medium

Find the escape velocity from the Moon:

GM = 4.9\times10^{12}

r = 1.74\times10^6 \text{ m}

Example 10

medium

A projectile is launched straight up at

8000 \text{ m/s}

from Earth (escape speed

11200 \text{ m/s}

). Will it escape?

Example 11

medium

Find the minimum kinetic energy to escape for a

500 \text{ kg}

probe where

GM = 4\times10^{14}

r = 6.4\times10^6 \text{ m}

Example 12

medium

A body has escape velocity

v_e

. What launch speed gives the object a speed-at-infinity equal to

v_e

? Use energy conservation.

Example 13

medium

Compare the escape velocities of two planets: planet A (

M

R

) and planet B (

2M

2R

Example 14

medium

A rocket reaches

11200 \text{ m/s}

(Earth's escape speed) at the surface. Ignoring air resistance, what is its speed very far away?

Example 15

medium

Find the escape velocity from the Sun's surface:

GM_{Sun} = 1.33\times10^{20}

r = 7\times10^8 \text{ m}

Example 16

medium

Find the escape velocity from a planet with

GM = 1 imes10^{14}

and radius

r = 5 imes10^6 ext{ m}

Example 17

medium

A planet has escape velocity

20 ext{ km/s}

. Find the orbital speed for a low circular orbit at its surface.

Example 18

challenge

Derive escape velocity by setting total energy to zero:

\frac{1}{2}mv_e^2 - \frac{GMm}{r} = 0

. Solve for

v_e

Example 19

challenge

A black hole has escape velocity equal to the speed of light

c = 3\times10^8 \text{ m/s}

at its radius. Find

r

in terms of

GM

(the Schwarzschild radius from this model).

Example 20

challenge

A probe is launched at

1.5

times Earth's escape velocity. Find its speed at infinity in terms of

v_e

Example 21

easy

If the radius

r

doubles (mass fixed), escape velocity changes by what factor?

Example 22

easy

If the mass

M

doubles (radius fixed), escape velocity changes by what factor?

Example 23

easy

Does a

1\,\text{kg}

object require the same escape speed from Earth as a

10^6\,\text{kg}

rocket (ignoring air)?

Example 24

easy

A satellite is in a low circular orbit at speed

v_{orb}

. Multiplying

v_{orb}

by what factor makes it escape?

Example 25

easy

After escape, does the object's speed approach zero, a constant non-zero value, or grow without bound far from the body?

Example 26

easy

Mars has

GM \approx 4.28\times 10^{13}

and

r \approx 3.39\times 10^6 \text{ m}

. Estimate its escape velocity.

Example 27

medium

A planet has surface gravity

g

and radius

R

. Express

v_e

in terms of

g

and

R

Example 28

medium

Using

v_e = \sqrt{2 g R}

, estimate Earth's escape velocity with

g = 9.8

m/s

^2

and

R = 6.4\times 10^6

Example 29

medium

Jupiter has

g \approx 24.8

m/s

^2

at radius

r \approx 6.99\times 10^7

m. Compute its escape velocity.

Example 30

medium

1000\,\text{kg}

probe must escape Earth. Find the minimum kinetic energy required at the surface (

v_e = 11200

m/s).

Example 31

medium

Body A has twice Earth's mass and the same radius. Body B has the same mass as Earth but half the radius. By what factor does each body's surface escape velocity exceed Earth's?

Example 32

medium

A planet's escape velocity is

15\,\text{km/s}

. Find its low-orbit circular speed.

Example 33

medium

A body launched at exactly Earth's escape speed reaches

r = 4R_E

. Compare its speed there to Earth's escape speed.

Example 34

medium

50\,\text{kg}

object escapes from a body with

v_e = 8\,\text{km/s}

. Find its kinetic energy at launch.

Example 35

medium

A small asteroid has

g_{surface} = 0.01

m/s

^2

and radius

r = 1000

m. Estimate its escape velocity.

Example 36

hard

Find escape velocity from Earth's surface (

v_{e0} = 11.2

km/s) at altitude

h = R_E

above the surface (

r = 2 R_E

Example 37

hard

A probe is launched from Earth at

v_0 = 12.5\,\text{km/s}

, surface escape speed

v_e = 11.2\,\text{km/s}

. Find its speed at infinity.

Example 38

hard

A planet has the same composition (density) as Earth and three times the radius. Find its escape velocity relative to Earth's.

Example 39

hard

From the Sun's surface (escape

v_e \approx 618

km/s), at what radius does escape speed drop to Earth's orbital speed

\sim 30

km/s?

Example 40

hard

Show that the total mechanical energy of an object launched at escape speed from a body is exactly zero at all distances.

Example 41

hard

If a black hole has mass equal to the Sun (

GM \approx 1.33\times 10^{20}

), compute its Schwarzschild radius using

v_e = c

Example 42

hard

A projectile launched from a planet of escape speed

v_e

at speed

v_0 < v_e

rises to maximum height

h

above the surface. Find

h

in terms of

R

v_0

, and

v_e

Example 43

hard

A planet has

g = 5

m/s

^2

and escape velocity

7\,\text{km/s}

. Find its radius.

Example 44

challenge

If Earth were compressed to half its radius without changing its mass, by what factor would its escape velocity change?

Example 45

challenge

A spacecraft escapes Earth (

v_{e,\oplus} = 11.2

km/s) and then must reach interstellar space, escaping the Sun's gravity at Earth's orbit (

v_{e,\odot} \approx 42.1

km/s). Find the total launch speed from Earth's surface needed (ignoring Earth's orbital boost).

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Related Concepts

Gravitational Field Gravitational Potential Energy

Background Knowledge

These ideas may be useful before you work through the harder examples.

gravitational fieldgravitational pe