Where does Newton’s 3rd Law come from?
How to prove “Every action has an equal and opposite reaction” is a direct result of conservation of momentum.
When I set out on my physics journey as a bright-eyed, I-will-make-a-groundbreaking-discovery-in-no-time hopeful, I was very pleased with myself when I could derive the principle of conservation of momentum from Newton’s 3rd Law.
However, I was soon knocked off my Dunning-Kruger confidence peak when I found out that conservation of momentum is universal whereas Newton’s 3rd Law only worked under certain conditions.
In essence, conservation of momentum is more fundamental than Newton’s 3rd Law.
It turns out I had proved things the wrong way around!
Many others like me will also have had this misconception because of the way most introductory physics courses cover forces before momentum.
This time, I will seek redemption from the higher-dimension physics deities by deriving it the right way around.
What is momentum and how does it relate to force?
We can think of momentum as ‘the amount of motion’ of an object or ‘how difficult is it to stop’. Officially, momentum, p, is the product of an object’s mass and velocity, p = mv.
This makes intuitive sense because an 18-tonne 16-wheeler cruising the road at 40 mph would be much more difficult (and dangerous!) to stop than a cyclist going at the same speed. So the greater mass 16-wheeler should have the greater ‘amount of motion’ or momentum.
The definition of momentum seems to draw a parallel with the most famous equation in physics (or perhaps second most after Einstein’s E = mc²):
F = ma (Newton’s 2nd Law)
p=mv
Because acceleration, a, is the rate of change of velocity, v, with respect to time,
assuming constant mass, m, force can be written as
F = m x rate of change of v, which is just
F = rate of change of p
Force is equal to the rate of change of momentum with respect to time.
(This is actually the general version of Newton’s 2nd Law. The familiar
F = ma only works for constant masses.)
Writing in terms of derivatives,
F = dp / dt
Principle of conservation of momentum
The total momentum of an isolated system remains constant.
(“Isolated” means there are no forces on the system by anything outside of it.)
Let’s unpack that statement with a simple example of two objects A and B, with initial momentums p₁ and p₂ respectively, in a head-on collision.
Conservation of momentum tells us that the vector sum
p₁+ p₂ = p₁’+ p₂’ = constant.
That is, the total momentum is not changing with time.
d (p₁+ p₂) / dt = 0.
d p₁ / dt + d p₂ / dt = 0
From Newton’s 2nd Law, F = dp / dt :
F₁ + F₂ = 0
F₁ = -F₂
“The force exerted on A by B is equal and opposite to the force exerted on B by A.”
Why is momentum conserved?
Trick question! It is actually a result of Noether’s Theorem, one of the most beautiful and fundamental theorems in physics. It is the basis for all conservation laws from momentum, angular momentum and energy to electric charge.
For each symmetry in the laws of nature, there is a corresponding conserved quantity.
(I will be not be deriving anything mathematical from Noether’s Theorem here but if you would like to explore, Physics with Elliot has made a great video on how it works.)
Essentially, momentum conservation is based on the idea that if you take an experiment, say a collision between two blocks, and move it to another room, you would get the same results.
It is said to have a translational symmetry in space. By Noether’s Theorem, the conserved quantity is momentum.
