Sep 4, 2015

$F = m \cdot a$

#### Explanation:

$\text{Force" = "mass" xx "acceleration}$

Rate of change of momentum is directly proportional to force.
Change in momentum can be

$m \cdot {v}_{2} - m \cdot {v}_{1} = m \cdot \Delta v$, where ${v}_{2}$ and ${v}_{1}$ are different velocities.

Rate of change of momentum is

$\frac{d}{\mathrm{dt}} \left(m v\right) = m \cdot \frac{\mathrm{dv}}{\mathrm{dt}}$

But $\frac{\mathrm{dv}}{\mathrm{dt}}$ is acceleration, so

$F = m \cdot a$

Sep 4, 2015

$\vec{F} = m \vec{a}$

#### Explanation:

Newton's he second law of motion describes how the motion of an object behaves under the influence of an external force. Here $\vec{F}$ is the total force on an object, this is a vector, since it has both a direction and a magnitude, $m$ the mass of the object and $\vec{a}$ the acceleration of the object, which is also a vector.

The law can also be written in a couple of different ways. since the acceleration is the change in velocity, and velocity the change in place, we can write $\vec{a} = \frac{d}{\mathrm{dt}} \vec{v} = \frac{d}{\mathrm{dt}} \frac{d}{\mathrm{dt}} \vec{x} = {d}^{2} / {\mathrm{dt}}^{2} \vec{x}$.
This means $\vec{F} = m \frac{d}{\mathrm{dt}} \vec{v} = {d}^{2} / {\mathrm{dt}}^{2} \vec{x}$.

Since the total mass of an object is usually constant, we can pull $m$ through the derivation, meaning $\vec{F} = \frac{d \left(m \vec{v}\right)}{\mathrm{dt}}$. Since we have the momentum of an object $\vec{p} = m \vec{v}$, we can state
$\vec{F} = \frac{\mathrm{dv} e c \left(p\right)}{\mathrm{dt}}$.
This can be used to show that in an isolated system the total momentum of all objects is conserved, which is an important result in classical physics.