How Do You Calculate Escape Velocity? (HL)

For IB Physics HL, calculate the escape speed from a distance $r$ from a spherical mass $M$ using $v_{\text{esc}}=\sqrt{\frac{2GM}{r}}$. From a planet’s surface, substitute its radius $R$ for $r$.

Here, $r$ is measured from the centre of the spherical body, not from its surface. The formula applies when the body's mass distribution is spherically symmetric.

Reasoning and derivation

Escape speed is the minimum initial speed needed for an object to reach infinity with zero final speed, assuming no propulsion or resistive forces. It is derived using conservation of energy.

At distance $r$, the object has kinetic energy and gravitational potential energy:

$$E=\frac{1}{2}mv^2-\frac{GMm}{r}$$

For the minimum escape speed, the object reaches infinity with zero kinetic energy. Gravitational potential energy is also zero at infinity, so the final total energy is zero:

$$\frac{1}{2}mv_{\text{esc}}^2-\frac{GMm}{r}=0$$

Rearranging gives:

$$v_{\text{esc}}=\sqrt{\frac{2GM}{r}}$$

The escaping mass $m$ cancels, so escape speed does not depend on the mass of the launched object.

For comparison, the circular orbital speed at the same radius is $v_{\text{orb}}=\sqrt{GM/r}$, so . This comparison applies only at the same value of .

Worked example: Earth

Using $G=6.67\times10^{-11}\ \text{N m}^2\text{ kg}^{-2}$, , and :

$$v_{\text{esc}}=\sqrt{\frac{2(6.67\times10^{-11})(5.97\times10^{24})}{6.37\times10^6}} =1.12\times10^4\ \text{m s}^{-1}$$

Therefore, Earth’s escape speed is approximately $11.2\ \text{km s}^{-1}$.

Model assumptions

The derivation treats the planet as isolated and spherical, the object as a point mass, and mechanical energy as conserved. It neglects atmospheric drag, propulsion after launch, and planetary rotation. A real launch therefore requires a different surface speed and trajectory.

A common misconception is that gravity becomes zero after escape speed is reached. Gravity continues acting; escape speed simply gives the object enough total mechanical energy not to return. This model also neglects atmospheric resistance and planetary rotation.

Exam technique: This is AHL content in D.1 Gravitational fields. For Calculate, show substitution and working. For Show, begin from total energy and include the negative gravitational potential energy term; omitting its minus sign is a common mark-losing error.