# Planck momentum

This plot of kinetic energy versus momentum has a place for most moving objects encountered in everyday life. It shows objects with the same kinetic energy (horizontally related) that carry different amounts of momentum, as well as how the speed of a low-mass object compares (by vertical extrapolation) to the speed after perfectly inelastic collision with a large object at rest. Highly sloped lines (rise/run=2) mark contours of constant mass, while lines of unit slope mark contours of constant speed. The plot further illustrates where lightspeed, Planck's constant, and kT figure in. (Note: the line labeled universe only tracks a mass estimate for the visible universe.)

Planck momentum is the unit of momentum in the system of natural units known as Planck units. It has no commonly used symbol of its own, but can be denoted by ${\displaystyle m_{\text{P}}c}$, where ${\displaystyle {m_{\text{P}}}}$ is the Planck mass and ${\displaystyle c}$ is the speed of light in a vacuum. Then

${\displaystyle m_{\text{P}}c={\frac {\hbar }{l_{\text{P}}}}={\sqrt {\frac {\hbar c^{3}}{G}}}\approx 6.52485{\text{ kg m/s}},}$

where

${\displaystyle \hbar }$ is the reduced Planck's constant,
${\displaystyle {l_{\text{P}}}}$ is the Planck length,
${\displaystyle G}$ is the gravitational constant.

In SI units Planck momentum is approximately 6.5 kg·m/s. It is equal to the Planck mass multiplied by the speed of light, usually associated with the momentum of primordial photons in some prevailing Big Bang models. Unlike most of the other Planck units, Planck momentum occurs on a human scale. By comparison, running with a five-pound object (108×Planck mass) at an average running speed (10−8×speed of light in a vacuum) would give the object Planck momentum. A 70 kg human moving at an average walking speed of 1.4 m/s (5.0 km/h; 3.1 mph) would have a momentum of about 15 ${\displaystyle m_{\text{P}}c}$. A baseball, which has mass ${\displaystyle m=}$ 0.145 kg, travelling at 45 m/s (160 km/h; 100 mph) would have a Planck momentum.