SME
Grade 11 Science · Problem type
A satellite in orbit
Use this for “A satellite circles Earth (5.97 × 10²⁴ kg) 7 × 10⁶ m from its center. How fast does it go, and how long is one orbit?”
Diagram
Orbital
speed:
v
=
√
GM/r
=
√(6.67
×
10
-11
×
5.97 ×
10
24
/(7
×
10
6
))
=
7,540
m/s
Period: T = 2πr/v = 2π × 7 × 10⁶/7,540 = 5,830 s (97.2 min)
Gravity is the centripetal force: GMm/r² = mv²/r, so the satellite’s own mass cancels.
It orbits 629,000 m above the surface (r − R, with R = 6.37 × 10⁶ m).
This is an example. Type your own values to start a new problem.
M
Central mass
example
r
Orbit radius
example
v
Orbital speed
calculated
T
Period
calculated · s
Formulas
v
=
√
6.674 ×
10
-11
×
M
r
7,540
=
√(6.674
×
10
-11
×
(5.97
×
10
24
)/(7
×
10
6
))
T
=
2π
×
r
v
5,830
=
2π
×
(7
×
10
6
)/7,540
Assumptions
•
The orbit is a circle, and gravity is the only force: it supplies the centripetal force.
•
GMm/r² = mv²/r, so v = √(GM/r), with G = 6.674 × 10⁻¹¹ N·m²/kg²; the satellite’s mass cancels.
•
r is measured from the center of the planet, not from its surface.
•
The orbit must be above the surface: for Earth, r is more than 6.37 × 10⁶ m.
•
T² = 4π²r³/(GM): doubling r makes T 2.83 times as long (Kepler’s third law).
Step-by-step
WE KNOW
M = 5.97 × 10²⁴ kg, r = 7 × 10⁶ m
FIND
orbital speed (v), period (T)
Step 1 · Find orbital speed (v)
Use v = √(6.674 × 10⁻¹¹ × M/r)
G
times
the
central
mass,
over
r
,
then
the
square
root.
v
=
√(6.674
×
10
-11
×
(5.97
×
10
24
)/(7
×
10
6
))
v
=
√((3.984
×
10
14
)/(7
×
10
6
))
v
=
√
5.691 ×
10
7
v
=
7,544.5
m/s
Step 2 · Find period (T)
Use T = 2π × r/v
Once
round
the
circle,
2πr,
at
speed
v
.
T
=
2π
×
(7
×
10
6
)/7,544.5
T
=
5,830
s
Check
Put the numbers back in. Both sides should match.
7,544.5
≈
√(6.674
×
10
-11
×
(5.97
×
10
24
)/(7
×
10
6
))
✓
5,830
≈
2π
×
(7
×
10
6
)/7,544.5
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Refresh
Related lessons
Circular motion and universal gravitation
Main lesson
A car rounding a flat curve
Use this for “A 1,200 kg car takes a 50 m curve at 15 m/s. How much friction does it need, and what is the least μₛ that holds it?”
At the bottom of a swing
Use this for “A 2 kg ball on a 1.5 m string swings through the bottom at 3 m/s. What is the tension there?”
Universal gravitation
Use this for “How hard do Earth (5.97 × 10²⁴ kg) and the Moon (7.35 × 10²² kg), 3.84 × 10⁸ m apart, pull on each other? What if they were twice as far?”