Orbital Motion Examples: 45 Problems with Answers

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

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

Concept Recap

Motion of an object held by gravity in a curved path — often a closed path (ellipse or circle) — around a planet, moon, star, or other mass.

An orbit is like falling around a planet instead of straight down onto it.

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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: Orbital Motion 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 orbital motion 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

A satellite orbits Earth (

GM = 4\times10^{14}

) at radius

r = 8\times10^6\text{ m}

. Find the period.

Answer

T \approx 7113 s

First step

v = \sqrt{GM/r} = \sqrt{5\times10^7} \approx 7071\text{ m/s}

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Example 2

medium

Two satellites orbit Earth at

r_1 = 7\times10^6\text{ m}

and

r_2 = 28\times10^6\text{ m}

. Find the ratio of their periods.

Example 3

medium

For a

500\text{ kg}

satellite at

r = 7\times10^6\text{ m}

around Earth (

GM = 4\times10^{14}

), find total mechanical energy.

Example 4

hard

A satellite drops from a circular orbit at radius

r_1 = 8\times10^6\text{ m}

r_2 = 7\times10^6\text{ m}

. Find the change in orbital speed (

GM = 4\times10^{14}

Example 5

hard

200\text{ kg}

satellite at

r = 7\times10^6\text{ m}

(

GM = 4\times10^{14}

) is boosted to a higher circular orbit at

r = 9\times10^6\text{ m}

. Find the work the engines must do.

Example 6

challenge

An elliptical orbit has perihelion

r_p = 6\times10^6\text{ m}

and aphelion

r_a = 1.4\times10^7\text{ m}

. Find the ratio of speeds

v_p/v_a

at these points.

Practice Problems

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

Example 1

easy

For a circular orbit, gravity provides what kind of force on the orbiting object?

In a circular orbit, gravity points toward the centre and acts as the centripetal force.

Example 2

easy

Find the orbital speed at radius

r=4\times10^7 \text{ m}

around a body with

GM = 4\times10^{14}

Example 3

easy

A satellite orbits at speed

v = 7000 \text{ m/s}

at radius

r = 7\times10^6 \text{ m}

. Find the centripetal acceleration.

Orbital velocity v is tangential; centripetal acceleration a points inward toward the planet. Find a.

Example 4

easy

Do lower orbits require higher or lower orbital speed than higher orbits?

Orbital speed is higher at smaller radius: v1 > v2 when r1 < r2.

Example 5

easy

Astronauts in orbit feel weightless. Is gravity zero there?

Example 6

easy

A satellite's orbital speed is

v = \sqrt{GM/r}

. If

r

quadruples, how does

v

change?

Example 7

easy

Find the orbital speed for a low Earth orbit where

GM = 4\times10^{14}

and

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

Example 8

easy

In orbit, the gravitational force on a satellite equals what expression for circular motion?

Setting F_g = F_c (gravity equals centripetal requirement) gives GMm/r² = mv²/r.

Example 9

medium

Find the orbital period for a circular orbit of speed

v = 8000 \text{ m/s}

at radius

r = 7\times10^6 \text{ m}

Example 10

medium

A satellite orbits at

r = 1.0\times10^7 \text{ m}

around Earth (

GM = 4\times10^{14}

). Find its orbital speed.

Example 11

medium

A geostationary orbit has period

T = 86400 \text{ s}

and

GM = 4\times10^{14}

. Find its radius using

r = (GM T^2/4\pi^2)^{1/3}

Example 12

medium

Two satellites orbit the same planet at radii

r

and

4r

. Find the ratio of their orbital speeds (inner to outer).

At radii r and 4r, the inner satellite's speed is twice the outer: v1 : v2 = 2 : 1.

Example 13

medium

A satellite of mass

200 \text{ kg}

orbits where

g = 8 \text{ N/kg}

at radius

r = 7\times10^6 \text{ m}

. Find its orbital speed using

g = v^2/r

The local field g supplies the centripetal acceleration. Since g = v²/r, the orbital speed v = √(gr).

Example 14

medium

A moon orbits a planet with period

T

. If a second moon orbits at

r

that is

4

times larger, find its period ratio using

T \propto r^{3/2}

Example 15

medium

Find the orbital speed of the Moon given

GM_{Earth} = 4\times10^{14}

and orbital radius

r = 3.84\times10^8 \text{ m}

Example 16

medium

A satellite orbits at radius

r = 8 imes10^6 ext{ m}

around Earth (

GM = 4 imes10^{14}

). Find its orbital speed.

Example 17

medium

A satellite has orbital speed

7500 ext{ m/s}

at radius

7 imes10^6 ext{ m}

. Find its orbital period.

Example 18

challenge

Show that for a circular orbit the kinetic energy is half the magnitude of the (negative) potential energy. Use

v^2 = GM/r

and

U = -GMm/r

Example 19

challenge

A satellite in circular orbit at

r

fires engines to double its speed instantaneously. Compare its new kinetic energy to the escape kinetic energy at that radius.

Doubling the orbital speed to 2v quadruples kinetic energy to 2× the escape KE — the satellite escapes.

Example 20

challenge

A satellite drops to a lower circular orbit (smaller

r

). Does its orbital speed increase or decrease, and what happens to its total energy?

Lower orbit → higher speed (v_lo > v_hi), yet total energy E

Example 21

easy

Find the orbital speed at radius

r = 1.6\times10^7\text{ m}

around a planet with

GM = 4\times10^{14}

Example 22

easy

A satellite orbits at

v = 5000\text{ m/s}

at radius

r = 1.0\times10^7\text{ m}

. Find its centripetal acceleration.

Example 23

easy

If a satellite's orbital radius is doubled, by what factor does its speed change (using

v \propto 1/\sqrt{r}

Example 24

easy

A satellite has orbital speed

7500\text{ m/s}

at radius

7\times10^6\text{ m}

. Find the gravitational acceleration on it.

Example 25

easy

Find the orbital speed at

r = 2\times10^7\text{ m}

around a planet with

GM = 4\times10^{14}

Example 26

medium

Use

T = 2\pi\sqrt{r^3/(GM)}

to find the period of a low Earth orbit at

r = 6.7\times10^6\text{ m}

(

GM = 4\times10^{14}

Example 27

medium

A satellite of mass

400\text{ kg}

orbits at

r = 7\times10^6\text{ m}

around Earth (

GM = 4\times10^{14}

). Find the gravitational force on it.

Example 28

medium

Find the orbital radius of a satellite around Earth (

GM = 4\times10^{14}

) with orbital speed

v = 6000\text{ m/s}

Example 29

medium

A satellite of mass

500\text{ kg}

orbits at radius

r = 7\times10^6\text{ m}

around Earth (

GM = 4\times10^{14}

). Find its kinetic energy.

Example 30

medium

A satellite at

r = 7\times10^6\text{ m}

has mass

500\text{ kg}

. Find its gravitational potential energy (

GM = 4\times10^{14}

Example 31

medium

A satellite orbits Mars (

GM_\text{Mars} = 4.28\times10^{13}

) at radius

r = 4\times10^6\text{ m}

. Find its orbital speed.

Example 32

medium

What is the escape speed from Earth's surface (

GM = 4\times10^{14}

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

Example 33

medium

A satellite orbits a planet with period

T = 1\text{ hr} = 3600\text{ s}

at radius

r = 7\times10^6\text{ m}

. Find the planet's

GM

Example 34

medium

An astronaut of mass

80\text{ kg}

orbits at

r = 7\times10^6\text{ m}

(

GM = 4\times10^{14}

). Find the gravitational force on her.

Example 35

hard

Find the radius of a geostationary orbit around a planet with

GM = 6.4\times10^{13}

and rotational period

T = 24\text{ hr} = 86400\text{ s}

Example 36

hard

100\text{ kg}

probe in circular orbit at

r = 1.0\times10^7\text{ m}

(

GM = 4\times10^{14}

) fires its engine, adding

1\times10^9\text{ J}

. Determine whether the probe escapes the planet.

Example 37

hard

A satellite at low Earth orbit has

T \approx 5500\text{ s}

. How many full orbits does it complete per day?

Example 38

hard

Two satellites are in circular orbits of radii

r

and

2r

around the same planet. Find the ratio of their kinetic energies (same mass).

Example 39

challenge

An elliptical orbit has semi-major axis

a = 1.0\times10^7\text{ m}

around Earth (

GM = 4\times10^{14}

). Find the orbital period.

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Related Concepts

Gravity Gravitational Field Centripetal Force Escape Velocity

Background Knowledge

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

gravitygravitational fieldcentripetal force