Escape Velocity needed to break free from a celestial body's gravitational field.

maths/power_using_iteration.py

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+def power(base: float, exponent: int) -> float:
+    """
+    Optimized power function using exponentiation by squaring.
+
+    Args:
+        base (float): The base number.
+        exponent (int): The exponent.
+
+    Returns:
+        float: The result of base raised to the power of exponent.
+
+    Examples:
+        >>> power(2, 3)
+        8.0
+        >>> power(5, -2)
+        0.04
+        >>> power(10, 0)
+        1.0
+        >>> power(7, 2)
+        49.0
+        >>> power(2, -3)
+        0.125
+        >>> power(2.5, 4)
+        39.0625
+        >>> power(-3.5, 2)
+        12.25
+        >>> power(-2, 3)
+        -8.0
+        >>> power(0, 5)
+        0.0
+        >>> power(0, 1)
+        0.0
+        >>> power(0, -1)
+        Traceback (most recent call last):
+            ...
+        ZeroDivisionError: 0.0 cannot be raised to a negative power.
+        >>> power(0, 0)
+        Traceback (most recent call last):
+            ...
+        ValueError: 0.0 raised to the power of 0 is indeterminate.
+        >>> power(1, 1000)
+        1.0
+
+    """
+    if base == 0 and exponent == 0:
+        raise ValueError("0.0 raised to the power of 0 is indeterminate.")
+    if base == 0 and exponent < 0:
+        raise ZeroDivisionError("0.0 cannot be raised to a negative power.")
+
+    result = 1.0
+    if exponent < 0:
+        base = 1 / base
+        exponent = -exponent
+    while exponent:
+        if exponent % 2 == 1:
+            result *= base
+        base *= base  # Square the base
+        exponent //= 2  # Halve the exponent
+    return round(result, 5)  # Round to 5 decimal places
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()
+    print("Raise base to the power of exponent using an optimized approach...")
+
+    try:
+        # Input handling and validation
+        base = float(input("Enter the base: ").strip())  # Supports float & int
+        exponent = int(
+            input("Enter the exponent: ").strip()
+        )  # Ensures exponent is an integer
+
+        # Calculate result
+        result = power(base, exponent)
+
+        # Display the result
+        print(f"{base} to the power of {exponent} is {result}")
+
+    except ValueError as e:
+        # Handle invalid input or indeterminate cases
+        print(e)
+    except ZeroDivisionError as e:
+        # Handle division by zero
+        print(e)

physics/escape_velocity.py

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+import math
+
+
+def escape_velocity(mass: float, radius: float) -> float:
+    """
+    Calculates the escape velocity needed to break free from a celestial body's gravitational field.
+
+    The formula used is:
+        v = sqrt(2 * G * M / R)
+    where:
+        v = escape velocity (m/s)
+        G = gravitational constant (6.67430 × 10^-11 m^3 kg^-1 s^-2)
+        M = mass of the celestial body (kg)
+        R = radius from the center of mass (m)
+
+    Args:
+        mass (float): Mass of the celestial body in kilograms.
+        radius (float): Radius from the center of mass in meters.
+
+    Returns:
+        float: Escape velocity in meters per second, rounded to 3 decimal places.
+
+    Examples:
+        >>> escape_velocity(5.972e24, 6.371e6)  # Earth
+        11185.978
+        >>> escape_velocity(7.348e22, 1.737e6)  # Moon
+        2376.307
+        >>> escape_velocity(1.898e27, 6.9911e7)  # Jupiter
+        60199.545
+        >>> escape_velocity(0, 1.0)
+        0.0
+        >>> escape_velocity(1.0, 0)
+        Traceback (most recent call last):
+            ...
+        ZeroDivisionError: Radius cannot be zero.
+    """
+    G = 6.67430e-11  # Gravitational constant in m^3 kg^-1 s^-2
+
+    if radius == 0:
+        raise ZeroDivisionError("Radius cannot be zero.")
+    if mass == 0:
+        return 0.0
+
+    velocity = math.sqrt(2 * G * mass / radius)
+    return round(velocity, 3)
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()
+    print("Calculate escape velocity of a celestial body...\n")
+
+    try:
+        # User input
+        mass = float(input("Enter mass of the celestial body (in kg): ").strip())
+        radius = float(input("Enter radius from center (in meters): ").strip())
+
+        # Result
+        velocity = escape_velocity(mass, radius)
+        print(f"Escape velocity is {velocity} m/s")
+
+    except ValueError:
+        print("Invalid input. Please enter valid numeric values.")
+    except ZeroDivisionError as e:
+        print(e)