 # Prove that F = ma.

F = ma is the formula of Newton’s Second Law of Motion.

Newton’s Second Law of Motion is defined as Force is equal to the rate of change of momentum. For a constant mass, force equals mass times acceleration.

### Newton’s second law

The second law of Newton can be described simply as the acceleration of an entity as produced by a net force is directly proportional to the magnitude of the net force, in the same path as the net force, and inversely proportional to the mass of the entity.

### Proof for F=ma

Let us consider an object of mass m, moving along a straight line with an initial velocity u. Let us say, after a certain time t, with a constant acceleration, the final velocity becomes v. Here we see that, the initial momentum

p1 = m × u

The final momentum

p2 = m × v

The change in momentum can be expressed as

p2 – p1 = (m × v) – (m × u)

p2 – p1 = m (v – u)

But we know that the rate of change of momentum with respect to time is proportional to the applied force.

The applied force

F ∝ [m (v – u)]/t

F ∝ m × a

as acceleration (a) = rate of change of velocity with respect to time.

F = k × m × a

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