Escape Velocity and Black Holes

Escape Velocity and Black Holes

The escape velocity is the minimum speed an object needs to escape a gravitational field entirely — travelling to infinity with zero remaining kinetic energy.

Escape Velocity Formula

Setting kinetic energy equal to gravitational potential energy:

$v_{\rm esc} = \sqrt{\frac{2GM}{R}}$

The Schwarzschild Radius

A black hole forms when an object is compressed below its Schwarzschild radius, where even light cannot escape:

$r_s = \frac{2GM}{c^2}$

For the Sun, $r_s \approx 2954$ m — about 3 km. For Earth, $r_s \approx 9$ mm.

Surface Gravity

The gravitational acceleration at a body's surface:

$g = \frac{GM}{R^2}$

Earth's surface gravity is $g \approx 9.82$ m/s².

Your Task

Implement three functions. Use $G = 6.674 \times 10^{-11}$ m³ kg⁻¹ s⁻² and $c = 2.998 \times 10^8$ m/s, defined inside each function.

• escape_velocity_m_s(M_kg, R_m) — escape velocity in m/s
• schwarzschild_radius_m(M_kg) — Schwarzschild radius in metres
• surface_gravity_m_s2(M_kg, R_m) — surface gravitational acceleration in m/s²