SME
University Physics I: Mechanics · Problem type
The rocket equation
Use this for “A 12,000 kg rocket burns fuel until it weighs 4,000 kg, its exhaust leaving at 3,000 m/s. How much speed does it gain?”
Chart
Δv(2,000) ≈ 2,291
This is an example. Type your own values to start a new problem.
Δv (m/s)
=
u (m/s)
×
ln
m₀ (kg)
mf (kg)
R
Mass ratio m₀ ÷ mf
worked out (not typed)
Formulas
R
=
m₀
mf
2.5
=
5,000
2,000
Δv
=
u
×
ln(
R
)
2,290.7268 = 2,500 × ln(2.5)
Assumptions
•
No gravity or air drag acts during the burn (a rocket in deep space), so the momentum of the rocket and its exhaust together is kept.
•
The exhaust leaves at a steady speed u measured from the rocket, straight backward.
•
Throwing back a small mass dm gives m dv = u dm; adding these up as the mass falls from m₀ to mf gives Δv = u ln(m₀ ÷ mf).
Step-by-step
WE KNOW
u = 2,500 m/s, m₀ = 5,000 kg, mf = 2,000 kg
FIND
mass ratio m₀ ÷ mf (R), change in speed (Δv)
Step 1 · Find mass ratio m₀ ÷ mf (R)
Use R = m₀ ÷ mf
The mass ratio is the start mass over the end mass: how many times lighter the rocket gets.
R
=
5,000
2,000
R
=
2.5
Step 2 · Find change in speed (Δv)
Use Δv = u × ln(R)
The speed gained is the exhaust speed times the natural log of the mass ratio.
Δv = 2,500 × ln(2.5)
Δv = 2,500 × 0.9163
Δv = 2,290.727 m/s
Check
Put the numbers back in. Both sides should match.
2.5
=
5,000
2,000
✓
2,290.727 = 2,500 × ln(2.5) ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Related lessons
Momentum
Main lesson
An elastic collision in 1D, both moving
Use this for “A 0.5 kg puck sliding east at 3 m/s catches a 1.5 kg puck sliding east at 1 m/s, and they bounce apart elastically. What are their velocities after?”
Center of mass of masses on a line
Use this for “A light 1.2 m rod holds 0.4 kg at 0.1 m, 0.6 kg at 0.5 m and 1 kg at 1.1 m. Where is its center of mass?”
Impulse from a triangular force pulse
Use this for “A kick pushes a 0.43 kg ball at rest with a force that rises evenly to 2,400 N and falls back to 0 over 8 ms. What is the impulse, and how fast does the ball leave?”