# Classical Physics from Newton's Laws?

## Main Question or Discussion Point

How much of classical physics can be purely derived from Newton's 3 Laws of motion?

Can Newton's laws derive conservation of momentum? Or any other conservation laws?

Consider this example

Let there be no external forces.

When standing in a stationary bus and the bus accelerates forwards, the person thrusts in the direction opposite the acceleration of the bus.

How do you explain this?

1. Newton’s third law, every action has an equal and opposite reaction. A 10N force exerted by the bus to the right means a 10N force on the person to the left. The person weighs less than the bus so accelerates more to the left than the bus to the right.

2. Conservation of linear momentum. The centre of mass of the initial system is stationary so centre of mass will stay in the one place as long as no external forces act. When the bus moves to the right, the person must move to the left in order to maintain the original position of the centre of mass. However the bus is much heavier so the person has to move faster to the left in order to maintain the position of the centre of mass. Hence the person accelerates quicker to the left.

Both explanations match observation although using different principles. So I wonder whether the two principles are linked, if not derivable from each other?

OR is it the case that classical physics is usually explanined from conservation laws plus Newton's Laws? So the conservation laws and Newton's laws are separate?

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Conservation of momentum and Newton's third law are equivalent. However, your example does not demonstrate either. Instead, it is a demonstration of inertia - the principle behind Newton's first law. Newton's third law applies only to the forces that two objects exert on each other. When a bus accelerates, it is due to the interaction between the bus and the ground, not that between the bus and its passengers.

Conservation of momentum and Newton's third law are equivalent.
How do you show this?

However, your example does not demonstrate either. Instead, it is a demonstration of inertia - the principle behind Newton's first law. Newton's third law applies only to the forces that two objects exert on each other. When a bus accelerates, it is due to the interaction between the bus and the ground, not that between the bus and its passengers.

How do you use Newton's first law to explain my example? I understand why the bus moves forwards but why does the passenger move backwards according to Newton's first law?

The passenger only apears to move backwards relative to the bus because he slips on the floor. If he was tied down like the seats the situation would be different.

The passenger only apears to move backwards relative to the bus because he slips on the floor. If he was tied down like the seats the situation would be different.
We assume no external forces other than the ground, bus and passenger.

dextercioby
Homework Helper
How do you show this?
It's a standard proof in any mechanics text. If you divide the forces acting on a system of particles in internal (pairs according to the III-rd postulate) forces and external forces (pairs as well, but we're interested only in those acting on the particles in the system), then the II-nd law says

$$\frac{d\vec{P}_{total, system}}{dt}=\sum (\vec{F}_{internal}+\vec{F}_{external})$$

Since by the III-rd postulate and the principle of forces' independence it follows that

$$\sum \vec{F}_{internal}=\vec{0}$$

then, the II-nd law becomes

$$\frac{d\vec{P}_{total, system}}{dt}=\vec{0}$$

if there are no external forces, or, more generally, if their vector sum is zero.

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One thing that has always amused me, in some sense, is that Newton's Third Law isn't generally correct. It fails for magnetism.

Yet somehow, if you calculate the momentum of the E and B fields, and the particles involved, it is still conserved. That's always bugged me a bit, although I'm glad it happens.