Acceleration Due to Gravity at a Distance

The Acceleration Due to Gravity at a Distance Calculator computes the gravitational acceleration exerted by an object of a specified mass at a given distance. This calculator uses the universal gravitational formula to determine the result, allowing for inputs in various mass and distance units and providing outputs in both metres per second squared (m/s²) and feet per second squared (ft/s²).

Note:

You can enter values in scientific notation in the following formats:
for large numbers: 5.972e24 
for small numbers: 1.67e-27

Calculator

1. enter the mass of the object and select the unit
2. enter the distance and select the unit
3. decimal precision, the number of digits after the decimal point

Results

1. the calculated gravitational acceleration in metres per second squared
2. metres per second squared in scientific notation
3. gravitational acceleration in feet per second squared
4. feet per second squared in scientific notation

Acceleration Due to Gravity

Formulas Used:

1. Acceleration Due to Gravity (\( a \)):

\[ a = \frac{G \cdot m}{d^2} \]

Where:

• \( a \): Acceleration in \(\text{m/s}^2\)
• \( G = 6.67430 \times 10^{-11} \, \text{m}^3/(\text{kg} \cdot \text{s}^2) \): Gravitational constant
• \( m \): Mass in kilograms
• \( d \): Distance in meters

2. Acceleration in Feet per Second Squared (\( a_{\text{ft/s}^2} \)):

\[ a_{\text{ft/s}^2} = a \cdot 3.28084 \]

Example Calculation:

Inputs:

• Mass (\( m \)): \( 5.972 \times 10^{24} \, \text{kg} \) (Earth's mass)
• Distance (\( d \)): \( 6.371 \times 10^6 \, \text{m} \) (Earth's radius)

Step 1: Calculate Acceleration in \(\text{m/s}^2\):

\[ a = \frac{G \cdot m}{d^2} \]

Substituting the values:

\[ a = \frac{(6.67430 \times 10^{-11}) \cdot (5.972 \times 10^{24})}{(6.371 \times 10^6)^2} \]

Numerator:

\[ (6.67430 \times 10^{-11}) \cdot (5.972 \times 10^{24}) = 3.986004418 \times 10^{14} \]

Denominator:

\[ (6.371 \times 10^6)^2 = 4.05841049 \times 10^{13} \]

Result:

\[ a = \frac{3.986004418 \times 10^{14}}{4.05841049 \times 10^{13}} = 9.798 \, \text{m/s}^2 \]

Step 2: Convert to \(\text{ft/s}^2\):

\[ a_{\text{ft/s}^2} = 9.798 \cdot 3.28084 = 32.087 \, \text{ft/s}^2 \]