# Question #b10d6

Apr 2, 2017

See below

#### Explanation:

Newton's 2nd Law, $\textcolor{b l u e}{F = m a}$ predicts that if the (net) force, $F$, on a body of mass $m$ is zero, then that body's acceleration (which is denoted $a$) is zero.

ie: $F = 0 \implies a = \frac{F}{m} = \frac{0}{m} = 0$

So, if a body is not accelerating, it will experience no change in velocity. It will therefore remain at rest - or continue to move at a constant velocity.

Newton's 1st Law states, roughly, that an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force.

So his 1st law looks to be a "mere" sub-set of his 2nd.

Seems odd but it may have been done this way to confront, head-on, Aristotle's idea that everything that is in motion is thus because there is some cause (ie a force) that is driving that motion. Who knows? Newton was hardly a team-player :)

Other thing worth stressing, I think, is that the 2nd Law has many manifestations. We start at:

$F = m a$

But it's really about the net force on an object, so:

$\sum F = m a$

And these are vectors , ie direction matters. A particle moving at constant speed but changing direction is accelerating. Eg uniform circular motion. So:

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

Interim calculus step:

$\sum \vec{F} = m \frac{d \vec{v}}{\mathrm{dt}}$

Really important where mass is not constant:

$\sum \vec{F} = \frac{d \left(m \vec{v}\right)}{\mathrm{dt}}$

Consolidating all of the above:

$\sum \vec{F} = \frac{d \vec{p}}{\mathrm{dt}}$

The third law is IMHO the truly interesting insight - it leads to universal conservation of momentum.