Training Aerospace Engineering Rocket Propulsion — Thrust & Δv
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Rocket Propulsion — Thrust & Δv

24 min Aerospace Engineering

Rocket Propulsion — Thrust & Δv

Rockets work by expelling mass at high velocity. The Tsiolkovsky rocket equation is the fundamental result relating fuel mass to velocity change.

Thrust

$$F = \dot{m} \, v_e$$

where $\dot{m}$ = mass flow rate (kg/s) and $v_e$ = effective exhaust velocity (m/s).

Tsiolkovsky Rocket Equation

$$\Delta v = v_e \ln\frac{m_0}{m_f}$$

where $m_0$ = initial mass (with fuel), $m_f$ = final mass (without fuel). The ratio $m_0/m_f$ is the mass ratio.

Specific Impulse

$$I_{sp} = \frac{v_e}{g_0}$$

Measured in seconds. Higher $I_{sp}$ means more efficient propulsion. $g_0 = 9.81$ m/s².

Example 1

A rocket engine has $I_{sp} = 300$ s and burns fuel at 200 kg/s. Find the thrust.

$v_e = I_{sp} \times g_0 = 300 \times 9.81 = 2{,}943$ m/s

$$F = 200 \times 2{,}943 = 588{,}600 \text{ N} \approx 589 \text{ kN}$$

Example 2

A spacecraft has $m_0 = 10{,}000$ kg, $m_f = 3{,}000$ kg, $I_{sp} = 350$ s. Find $\Delta v$.

$v_e = 350 \times 9.81 = 3{,}433.5$ m/s

$$\Delta v = 3{,}433.5 \times \ln\frac{10{,}000}{3{,}000} = 3{,}433.5 \times 1.204 = 4{,}134 \text{ m/s}$$

Example 3

What mass ratio is needed for $\Delta v = 9{,}400$ m/s (to reach LEO) with $I_{sp} = 310$ s?

$v_e = 310 \times 9.81 = 3{,}041$ m/s

$$\frac{m_0}{m_f} = e^{\Delta v / v_e} = e^{9400/3041} = e^{3.09} = 22.0$$

This means 95.5% of the initial mass must be fuel — hence the need for staging!

Practice Problems

1. A rocket engine: $\dot{m} = 500$ kg/s, $v_e = 3{,}500$ m/s. Find thrust.
2. Find $I_{sp}$ for $v_e = 4{,}500$ m/s.
3. $m_0 = 20{,}000$ kg, $m_f = 5{,}000$ kg, $I_{sp} = 280$ s. Find $\Delta v$.
4. What mass ratio gives $\Delta v = 3{,}000$ m/s with $I_{sp} = 320$ s?
5. A two-stage rocket: Stage 1 has mass ratio 4, $I_{sp} = 280$ s. Stage 2 has mass ratio 3, $I_{sp} = 340$ s. Find total $\Delta v$.
6. An ion thruster has $I_{sp} = 3{,}000$ s but $\dot{m} = 0.001$ kg/s. Find thrust.
7. Compare the fuel efficiency of engines with $I_{sp} = 250$ s vs. $I_{sp} = 450$ s.
8. A spacecraft needs $\Delta v = 2$ km/s for a maneuver. $I_{sp} = 300$ s, dry mass 500 kg. Find the fuel mass needed.
9. If burn time is 120 s and $\dot{m} = 100$ kg/s, find total fuel consumed and thrust.
10. Gravity loss: effective $\Delta v$ is reduced by $g \cdot t_{\text{burn}}$. If burn time is 150 s, find gravity loss.
11. The Saturn V first stage: $F = 33{,}000$ kN, $\dot{m} = 13{,}000$ kg/s. Find $v_e$ and $I_{sp}$.
12. Why is staging necessary for reaching orbit? Explain using the mass ratio.
Show Answer Key

1. $F = 500 \times 3{,}500 = 1{,}750$ kN

2. $I_{sp} = 4{,}500/9.81 = 458.7$ s

3. $v_e = 2{,}746.8$ m/s; $\Delta v = 2{,}746.8 \ln 4 = 3{,}806$ m/s

4. $v_e = 3{,}139$ m/s; $m_0/m_f = e^{3000/3139} = e^{0.956} = 2.60$

5. $\Delta v_1 = 280(9.81)\ln 4 = 3{,}806$ m/s; $\Delta v_2 = 340(9.81)\ln 3 = 3{,}666$ m/s; Total $= 7{,}472$ m/s

6. $F = 0.001 \times 3000 \times 9.81 = 29.4$ N (very low thrust, but very efficient)

7. $v_e$ ratio $= 450/250 = 1.8$; the higher $I_{sp}$ engine extracts $1.8\times$ more $\Delta v$ per kg of fuel

8. $m_0/m_f = e^{2000/2943} = e^{0.6795} = 1.973$; $m_0 = 986.5$ kg; fuel $= 486.5$ kg

9. Fuel $= 100 \times 120 = 12{,}000$ kg; need $v_e$ to find thrust — $F = \dot{m} v_e$

10. Gravity loss $= 9.81 \times 150 = 1{,}472$ m/s

11. $v_e = 33 \times 10^6/13{,}000 = 2{,}538$ m/s; $I_{sp} = 2{,}538/9.81 = 259$ s

12. Reaching LEO requires $\Delta v \approx 9.4$ km/s. A single stage needs mass ratio $\sim 20+$, meaning > 95% fuel. By staging, each stage has a manageable ratio (3–5), and empty tanks are jettisoned to reduce dead weight.

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