Derive relation F = ma from Newton 2nd Law of Motion

Let us derive the relation of force F = ma from Newton’s second law:

According to the Newton’s 2nd Law of motion, the rate of change of linear momentum of a body is directly proportional to the applied external force and in the direction of force.

It means that the linear momentum will change faster when a bigger force is applied.

Consider a body of mass ‘m’ moving with velocity v.

The linear momentum of a body is given by:

p = mv

Now According to Newton’s 2nd Law of Motion:

Force is directly proportional to rate of change of momnetum, that is

F α dp/dt

F  = k dp/dt

F = k d(mv)/dt

F = k md(v)/dt

F = k ma

Experimentally k =1

F = k ma

Which is the required equation of force.


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7 Replies to “Derive relation F = ma from Newton 2nd Law of Motion”

  1. it would be clearer if u mentioned p=mv and v/t=a and thus ma arises out,

    anyways good work and thanks.

  2. From F=ma, if we derive for acceleration it is, a=F/m…As per this the unit of acceleration would become Newton/Kilogram (N/kg)…
    But the actual unit of acceleration is m/s^2…
    I am confused…

    1. A Newton is a kgm/s^2. So a kgm/s^2/kg leaves a unit of m/s^2. You just have to break the Newton down into its components to cancel the kg and get the acceleration in typical units.

  3. I have been lurking though the textbook and guides to understand the derivation !
    This finally helped me ! Thanks a lot

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