Momentum Equation

What momentum means

A moving object has momentum. Momentum describes its motion by combining two properties: how much mass the object has and how fast it is moving in a particular direction. A heavy object and a fast object can therefore both have large momentum.

Momentum is defined by

$p=m\times v$

where:

• $p$ is momentum in kilogram metres per second (kg m/s)
• $m$ is mass in kilograms (kg)
• $v$ is velocity in metres per second (m/s).

The symbol $p$ is conventionally used for momentum. The equation also explains the unit:

$\text{kg}\times\text{m/s}=\text{kg m/s}$

Mass must be in kilograms and velocity must be in metres per second before the equation is used. A mass given in grams must first be divided by 1000 to convert it to kilograms.

How mass and velocity affect momentum

For a fixed velocity, increasing the mass increases the momentum in the same proportion. Doubling the mass doubles the momentum. This is why a heavy vehicle travelling at a given velocity has more momentum than a lighter vehicle travelling at that velocity.

For a fixed mass, increasing the magnitude of the velocity also increases the magnitude of the momentum in the same proportion. Doubling the velocity doubles the momentum. If an object is stationary, $v=0$, so

$p=m\times0=0.$

A stationary object therefore has zero momentum, whatever its mass.

The equation may also be rearranged:

$m=\frac{p}{v}$

and

$v=\frac{p}{m}.$

Use the form that places the required quantity by itself, then substitute values with consistent units.

Momentum has a direction

Momentum is a vector quantity because velocity is a vector. Its direction is the same as the object's velocity. Mass has no direction, but multiplying it by velocity does not remove velocity's direction.

For motion along a straight line, choose one direction as positive. For example, right may be positive, making left negative. An object moving right then has positive momentum, while an object moving left has negative momentum. The minus sign records direction; it does not mean that the object has a negative mass.

Changing direction changes momentum even if speed and mass stay the same. For example, reversing from a velocity of $+v$ to $-v$ reverses the momentum from $+mv$ to $-mv$. This directional property becomes important when momentum is used to describe collisions, but the essential starting point is always the same: identify the mass, use the velocity with its direction, and apply $p=mv$.

Core equation

$p=mv$

• $p$: momentum in kg m/s
• $m$: mass in kg