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# Cheat sheet

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Kerbal Space Program rocket scientist's cheat sheet: Delta-v maps, equations and more for your reference so you can get from here to there and back again.

## Mathematics

### Thrust to Weight Ratio (TWR)

This is Newton's Second Law. If the ratio is less than 1 the craft will not lift off the ground. Note that the local gravitational acceleration, which is usually the surface gravity of the body the rocket is starting from, is required.

$\text{TWR} = \frac{F_T}{m \cdot g} > 1$
Where:
• $F_T$ is the thrust of the engines
• $m$ the total mass of the craft
• $g$ the local gravitational acceleration (usually surface gravity)

### Combined Specific Impulse (Isp)

If the Isp is the same for all engines in a stage, then the Isp is equal to a single engine. If the Isp is different for engines in a single stage, then use the following equation:

$I_{sp} = \frac{(F_1 + F_2 + \dots)}{\frac{F_1}{I_{sp1}} + \frac{F_2}{I_{sp2}} + \dots}$

### Delta-v (Δv)

#### Basic calculation

Basic calculation of a rocket's Δv. Use the atmospheric and vacuum thrust values for atmospheric and vacuum Δv, respectively.

$\Delta{v} = ln\left(\frac{M_{start} }{M_{end} }\right) \cdot I_{sp} \cdot 9.81 \frac{m}{s^2}$
Where:
• $\Delta{v}$ is the velocity change possible in m/s
• $M_{start}$ is the starting mass in the same unit as $M_{end}$
• $M_{end}$ is the end mass in the same unit as $M_{start}$
• $I_{sp}$ is the specific impulse of the engine in seconds

#### True Δv of a stage that crosses from atmosphere to vacuum

Body Δvout
Kerbin 1000 m/s
other bodies' data missing

Calculation of a rocket stage's Δv, taking into account transitioning from atmosphere to vacuum. Δvout is the amount of Δv required to leave a body's atmosphere, not reach orbit. This equation is useful to figure out the actual Δv of a stage that transitions from atmosphere to vacuum.

$\Delta{v}_T = \frac{\Delta{v}_{atm} - \Delta{v}_{out}}{\Delta{v}_{atm}} \cdot \Delta{v}_{vac} + \Delta{v}_{out}$

#### Maps

Various fan-made maps showing the Δv required to travel to a certain body.

Subway style Δv map:

Total Δv values

Δv change values

Δv nomogram

Δv with Phase Angles

Precise Total Δv values

## Math examples

### TWR

• Copy template:
TWR = F / (m * g) > 1

### Isp

1. When Isp is the same for all engines in a stage, then the Isp is equal to a single engine. So six 200 Isp engines still yields only 200 Isp.
2. When Isp is different for engines in a single stage, then use the following equation:
• Equation:

$I_{sp} = \frac{(F_1 + F_2 + \dots)}{\frac{F_1}{I_{sp1}} + \frac{F_2}{I_{sp2}} + \dots}$

• Simplified:
Isp = ( F1 + F2 + ... ) / ( ( F1 / Isp1 ) + ( F2 / Isp2 ) + ... )
• Explained:
Isp = ( Force of Thrust of 1st Engine + Force of Thrust of 2nd Engine...and so on... ) / ( ( Force of Thrust of 1st Engine / Isp of 1st Engine ) + ( Force of Thrust of 2nd Engine / Isp of 2nd Engine ) + ...and so on... )
• Example:
Two engines, one rated 200 newtons and 120 seconds Isp ; another engine rated 50 newtons and 200 seconds Isp.
Isp = (200 newtons + 50 newtons) / ( ( 200 newtons / 120 ) + ( 50 newtons / 200 ) = 130.89 seconds Isp

### Δv

1. For atmospheric Δv value, use atmospheric thrust values.
2. For vacuum Δv value, use vacuum thrust values.
3. Use this equation to figure out the Δv per stage:
• Equation:

$\Delta{v} = ln\left(\frac{M_{start}}{M_{dry}}\right) \cdot I_{sp} \cdot 9.81 \frac{m}{s^2}$

• Simplified:
Δv = ln ( Mstart / Mdry ) * Isp * g
• Explained:
Δv = ln ( Starting Mass / Dry Mass ) X Isp X 9.81
• Example:
Single Stage Rocket that weighs 23 tons when full, 15 tons when fuel is emptied, and engine that outputs 120 seconds Isp.
Δv = ln ( 23 Tons / 15 Tons ) × 120 seconds Isp × 9.81m/s² = Total Δv of 503.2 m/s

### True Δv

1. How to calculate the Δv of a rocket stage that transitions from Kerbin atmosphere to vacuum.
2. Assumption: It takes approximately 1000 m/s of Δv to escape Kerbin's atmosphere before vacuum Δv values take over for the stage powering the transition.
3. Note: This equation is an guess, approximation, and is not 100% accurate. Per forum user stupid_chris who came up with the equation: "The results will vary a bit depending on your TWR and such, but it should usually be pretty darn accurate."
• Equation for Kerbin Atmospheric Escape:

$\Delta{v}_T = \frac{\Delta{v}_{atm} - \Delta{v}_{out}}{\Delta{v}_{atm}} \cdot \Delta{v}_{vac} + \Delta{v}_{out}$

• Simplified:
True Δv = ( ( Δv atm - 1000 ) / Δv atm ) * Δv vac + 1000
• Explained:
True Δv = ( ( Total Δv in atmosphere - 1000 m/s) / Total Δv in atmosphere ) X Total Δv in vacuum + 1000
• Example:
Single Stage with total atmospheric Δv of 5000 m/s, and rated 6000 Δv in vacuum.
Transitional Δv = ( ( 5000 Δv atm - 1000 Δv Required to escape Kerbin atmosphere ) / 5000 Δv atm ) X 6000 Δv vac + 1000 Δv Required to escape Kerbin atmosphere = Total Δv of 5800 m/s