Question

derive from one of the laws of motion the relation between the momentum produced in a body and the force applied on the body​

Answer 1

The fact that change in moment is equal to force times duration can be derived from Newton's Second Law of Motion (assuming that the net force on the object is constant.)

Step-by-step explanation:

Let
`m` denote the mass of an object. Let
`\Delta v` denote the change in the velocity of this object. The change in momentum of this object will be
`\Delta p = m\, \Delta v`.

Let
`F_{\text{net}}` denote the net force (resultant force) on the object. Let
`a` denote the acceleration of this object. By the law of motion,
`F_{\text{net}} = m\, a`, or equivalently,
`a = (F_{\text{net}} / m)`.

Assume that this net force is constant and is exerted on this object over a period of
`\Delta t`. With an acceleration of
`a = (F_{\text{net}} / m)`, the velocity of this object would have changed by:

`\begin{aligned}\Delta v &= a\, \Delta t \\ &= \left(\frac{F_{\text{net}}}{m}\right)\, (\Delta t) \\ &= \frac{F_{\text{net}}\, \Delta t}{m}\end{aligned}`.

Multiply both sides of this equation by the mass
`m` of the object to obtain:

`m\, \Delta v = F_{\text{net}}\, \Delta t`.

Note that
`m\, \Delta v` is equal to the change in momentum (
`\Delta p = m\, \Delta v`.) Therefore, if the net force
`F_{\text{net}}` on the object is constant, the change in the momentum
`\Delta p` over time period
`\Delta t` will be equal to
`F_{\text{net}}\, \Delta t` (net impulse on the object.)