State and prove Newton’s second law of motion in terms of momentum.

Newton’s Second Law of Motion in Terms of Momentum:

Newton’s Second Law of Motion can be expressed in terms of momentum as follows:

Statement: The rate of change of momentum of a body is directly proportional to the net external force acting on it and occurs in the direction of the force.

Mathematically, the law in terms of momentum can be written as:

F=dp/dt

Where:

  • F is the net external force acting on the body.
  • p is the momentum of the body.
  • dp/dt is the rate of change of momentum.

Proof of Newton’s Second Law in Terms of Momentum:

  1. Momentum Definition: Momentum (p) of an object is defined as the product of its mass () and velocity (v):

    p=mv

    Where:

    • is the mass of the object.
    • v is the velocity of the object.

  2. Differentiating Momentum with Respect to Time: To find the rate of change of momentum, we differentiate momentum p=mv with respect to time .

    dp/dt=d/dt(mv)

    Assuming the mass of the object is constant, we can apply the product rule:

    dp/dt=m.dv/dt

    The term dv/dt is the acceleration (a) of the object, so we have:

    dp/dt=ma

  3. Relating to Force: By Newton’s Second Law in its traditional form:

    F=ma

    Therefore, from the equation above, we can see that:

    dp/dt=F

    This shows that the rate of change of momentum (dp/dt) is equal to the net external force F.

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