# If the velocity of an object doubles, does its momentum double?

Jul 10, 2014

The linear momentum (also known as the quantity of motion), by definition, is a product of a mass (a scalar) by velocity (a vector) and is, therefore, a vector:

$P = m \cdot V$

Assuming that the velocity doubles (that is, the vector of velocity doubles in magnitude retaining the direction), the momentum doubles as well, that is, it doubles in magnitude retaining the direction.

In classical mechanics there is a law of conservation of momentum that, combined with the law of conservation of energy, helps, for example, to determine the movement of objects after collision if we know their movements before the collision.

Incidentally, since an acceleration is a derivative of a velocity by time

$a = \frac{\mathrm{dV}}{\mathrm{dt}}$

And considering the second Newton's law relating the force $F$, mass $m$ and acceleration $a$

$F = m \cdot a$

we can relate the force and momentum $P = m \cdot V$ as

$F = \frac{\mathrm{dP}}{\mathrm{dt}} = \frac{d \left(m \cdot V\right)}{\mathrm{dt}}$