Question

Verify the principle of conservation of linear momentum using newton's second law of motion only?

Answer 1

Newton's second law:

`F = ma`

where `F` is the force acting on an object, `m` is the mass of the object, and `a` is the object's acceleration.

We can rewrite this using the definition `a equiv (Deltav)/(Deltat)`, where `Delta v` is the change in velocity of the object over some time duration `Delta t`.

`F = m (Delta v)/(Delta t)`

which is equivalent to:

`F = (Delta p)/(Delta t)`

where `Delta p equiv m Delta v` is change in momentum.

We know that when two objects apply forces on each other, they are equal and opposite (which is also Newton's 3rd law, so you can't justify conservation of momentum without it).

`F_("a on b") = -F_("b on a")`

`(Delta p_a)/(Delta t) = -(Delta p_b)/(Delta t)`

`Delta p_a = - Delta p_b`

So the changes in momentum of object a and object b are equal and opposite. More explicity:

`Delta p_a + Delta p_b = 0`

This demonstrates conservation of momentum, since the sum of the momentum of the objects is always zero.