SME
University Physics I: Mechanics · Problem type
Energy to lift a satellite to its orbit height
Use this for “How much potential energy does a 1000 kg satellite gain going from the ground to 400 km?”
Chart
U(6,771) ≈ −58.86
Level U₀ = −62.56 meets the curve at R = 6,371
This is an example. Type your own values to start a new problem.
ΔU (
)
=
G
M (kg)
×
m (kg)
1
R (
)
−
1
r (
)
h
Height above the surface
example
U₀
Potential energy on the ground
worked out (not typed)
U
Potential energy at the height
worked out (not typed)
E
Orbital energy
worked out (not typed)
Formulas
r
=
R
+
h
6.771 ×
10
6
=
6.371 ×
10
6
+
(4
×
10
5
)
U₀
=
−6.674 ×
10
-11
×
M
×
m
R
−6.256 ×
10
10
=
−6.674 ×
10
-11
×
(5.972
×
10
24
)
×
1,000
6.371 ×
10
6
U
=
−6.674 ×
10
-11
×
M
×
m
r
−5.8864 ×
10
10
=
−6.674 ×
10
-11
×
(5.972
×
10
24
)
×
1,000
6.771 ×
10
6
ΔU
=
6.674 ×
10
-11
×
M
×
m
×
h
R
×
r
+3.6958
×
10
9
=
6.674 ×
10
-11
×
(5.972
×
10
24
)
×
1,000
×
4 ×
10
5
6.371 ×
10
6
× (6.771 ×
10
6
)
E
=
−6.674 ×
10
-11
×
M
×
m
2 ×
r
−2.9432 ×
10
10
=
−6.674 ×
10
-11
×
(5.972
×
10
24
)
×
1,000
2 × (6.771 ×
10
6
)
Formulas are worked in m, J. The step-by-step shows the conversions.
Assumptions
•
U = −GMm ÷ r, zero far away at infinity; r is measured from the planet’s center.
•
G = 6.674 × 10⁻¹¹ N·m²/kg²; Earth: M = 5.972 × 10²⁴ kg and R = 6371 km.
•
mgh is the limit for h much smaller than R; g weakens with height, so ΔU comes out under mgh.
•
E = −GMm ÷ (2r) is the kinetic plus potential energy once the satellite circles at r.
Step-by-step
WE KNOW
m = 1,000 kg, h = 400 km, M = 5.972 × 10²⁴ kg, R = 6,371 km
FIND
orbit radius (r), potential energy on the ground (U₀), potential energy at the height (U), energy gained going up (ΔU), orbital energy (E)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to m, J).
h = 400 km = 400,000 m (1 km = 1000 m)
R = 6,371 km = 6,371,000 m (1 km = 1000 m)
Step 2 · Find orbit radius (r)
Use r = R + h
Distances
in
U
are
from
the
planet’s
center,
so
add
its
radius
to
the
height.
r
=
6.371 ×
10
6
+
(4
×
10
5
)
r
=
6,771,000
m
Step 3 · Find potential energy on the ground (U₀)
Use U₀ = −6.674 × 10⁻¹¹ × M × m ÷ R
On
the
ground
r
=
R
.
The
energy
is
below
0:
work
must
be
done
to
pull
the
two
apart.
U₀
=
−6.674 ×
10
-11
×
(5.972
×
10
24
)
×
1,000
6.371 ×
10
6
U₀
=
(−3.9857
×
10
14
)
×
1,000
6.371 ×
10
6
U₀
=
−3.9857 ×
10
17
6.371 ×
10
6
U₀
=
−6.256 ×
10
10
J
Step 4 · Find potential energy at the height (U)
Use U = −6.674 × 10⁻¹¹ × M × m ÷ r
The
same
law
at
the
height:
r
is
larger,
so
U
is
closer
to
0,
that
is
higher.
U
=
−6.674 ×
10
-11
×
(5.972
×
10
24
)
×
1,000
6.771 ×
10
6
U
=
(−3.9857
×
10
14
)
×
1,000
6.771 ×
10
6
U
=
−3.9857 ×
10
17
6.771 ×
10
6
U
=
−5.8864 ×
10
10
J
Step 5 · Find energy gained going up (ΔU)
Use ΔU = 6.674 × 10⁻¹¹ × M × m × h ÷ (R × r)
U
−
U₀
=
GMm(
1
R
−
1
r
),
and
1
R
−
1
r
=
h
Rr
over
the
common
denominator.
ΔU
=
6.674 ×
10
-11
×
(5.972
×
10
24
)
×
1,000
×
4 ×
10
5
6.371 ×
10
6
× (6.771 ×
10
6
)
ΔU
=
(6.674
×
10
-11
)
×
(5.972
×
10
24
)
×
1,000
×
4 ×
10
5
4.3138 ×
10
13
ΔU
=
(3.9857
×
10
14
)
×
1,000
×
4 ×
10
5
4.3138 ×
10
13
ΔU
=
(3.9857
×
10
17
)
×
4 ×
10
5
4.3138 ×
10
13
ΔU
=
1.5943 ×
10
23
4.3138 ×
10
13
ΔU
=
3.6958 ×
10
9
J
Step 6 · Find orbital energy (E)
Use E = −6.674 × 10⁻¹¹ × M × m ÷ (2 × r)
In
orbit
1
2
mv²
=
GMm
2
r
,
half
the
size
of
U
;
adding
it
to
U
leaves
half
of
U
.
E
=
−6.674 ×
10
-11
×
(5.972
×
10
24
)
×
1,000
2 × (6.771 ×
10
6
)
E
=
(−6.674
×
10
-11
)
×
(5.972
×
10
24
)
×
1,000
1.3542 ×
10
7
E
=
(−3.9857
×
10
14
)
×
1,000
1.3542 ×
10
7
E
=
−3.9857 ×
10
17
1.3542 ×
10
7
E
=
−2.9432 ×
10
10
J
Step 7 · Convert the answers
Change each answer back to the unit you chose.
r = 6,771,000 m = 6,771 km (1 km = 1000 m)
U₀ = −6.256 × 10¹⁰ J = −62.5602 GJ (1 GJ = 10⁹ J)
U = −5.8864 × 10¹⁰ J = −58.8645 GJ (1 GJ = 10⁹ J)
ΔU = 3.6958 × 10⁹ J = +3.6958 GJ (1 GJ = 10⁹ J)
E = −2.9432 × 10¹⁰ J = −29.4322 GJ (1 GJ = 10⁹ J)
Check
Put the numbers back in. Both sides should match.
6.771 ×
10
6
=
6.371 ×
10
6
+
(4
×
10
5
)
✓
−6.256 ×
10
10
=
−6.674 ×
10
-11
×
(5.972
×
10
24
)
×
1,000
6.371 ×
10
6
✓
−5.8864 ×
10
10
=
−6.674 ×
10
-11
×
(5.972
×
10
24
)
×
1,000
6.771 ×
10
6
✓
+3.6958
×
10
9
=
6.674 ×
10
-11
×
(5.972
×
10
24
)
×
1,000
×
4 ×
10
5
6.371 ×
10
6
× (6.771 ×
10
6
)
✓
−2.9432 ×
10
10
=
−6.674 ×
10
-11
×
(5.972
×
10
24
)
×
1,000
2 × (6.771 ×
10
6
)
✓
Topic check
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