SME
Flight Mechanics, Stability & Control · Problem type
Jet range (Breguet)
Use this for “A jet cruises at 230 m/s with L/D = 16 and cₜ = 0.6 per hour. It starts the cruise at 600,000 N and burns 120,000 N of fuel. Find its range.”
Chart
R(1.25) ≈ 4,927
This is an example. Type your own values to start a new problem.
W₀/W₁
=
W₀ (N)
W₀ (N)
−
Wf (N)
R (km)
=
V (m/s)
cₜ (h⁻¹)
×
L/D
×
ln(
W₀/W₁
)
Formulas
W₀
/W₁
=
W₀
W₀
− Wf
1.25
=
600,000
600,000 − 120,000
R
=
(
V
cₜ
)
×
(L/D)
×
ln(
W₀
/W₁)
4,927,000
=
(
230
0.0001667
)
×
16
×
ln(1.25)
Formulas are worked in s⁻¹, m. The step-by-step shows the conversions.
Assumptions
•
Constant airspeed V and lift-to-drag ratio L/D through the cruise, as in a cruise-climb.
•
Thrust-specific fuel consumption cₜ is the fuel weight burned per second per unit of thrust.
•
W₀ is the weight at the start of cruise and W₁ at the end, after the fuel burned.
Step-by-step
WE KNOW
V = 230 m/s, cₜ = 0.6 h⁻¹, L/D = 16, W₀ = 600,000 N, Wf = 120,000 N
FIND
weight ratio (W₀/W₁), range (R)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to s⁻¹, m).
cₜ = 0.6 h⁻¹ = 0.0001667 s⁻¹ (1 s⁻¹ = 3600 h⁻¹)
Step 2 · Find weight ratio (W₀/W₁)
Use W₀/W₁ = W₀ ÷ (W₀ − Wf)
The
weight
at
the
end
of
cruise
is
W₁
=
W₀
−
Wf,
the
start
weight
less
the
fuel
burned.
W₀
/W₁
=
600,000
600,000 − 120,000
W₀
/W₁
=
600,000
480,000
W₀
/W₁
=
1.25
Step 3 · Find range (R)
Use R = (V ÷ cₜ) × (L/D) × ln(W₀/W₁)
With
dR
=
−(
V
(L/D)
cₜ W
)
dW
and
u
=
W
W₁
,
the
range
is
V
(L/D)
cₜ
times
the
integral
of
1
u
from
1
to
W₀
W₁
.
R
=
(
230
0.0001667
)
×
16
×
ln(1.25)
1.25
∫
1
(
1
u
) du
=
[ln|u|]
from
1
to
1.25
=
0.2231
R
=
4,927,009.613
m
Step 4 · Convert the answers
Change each answer back to the unit you chose.
R = 4,927,000 m = 4,927 km (1 km = 1000 m)
Check
Put the numbers back in. Both sides should match.
1.25
=
600,000
600,000 − 120,000
✓
4,927,000
=
(
230
0.00016666667
)
×
16
×
ln(1.25)
✓
Topic check
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