# Derivation of expression for force

My higher secondary book says that momentum[p] is equal to 'mv' where 'm' is mass and 'v' is velocity.Then it was also mentioned that the force[f] was dependent on the rate of change of momentum{[delta]p= m*[delta]v}. Though the book mentions various examples to support that momentum plays a major role in motion, it did not give examples to support that the force depends on the rate of change of momentum{ie.. force also depends on time in addition to momentum}. Could you please give me some experiences from real life to derive the expression for force?

As far as I've been able to find out, it comes from Newton's First Law: "every body persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed". The body has a constant momentum in this case and a force would change the momentum over time, so F = dp/dt. At some point, Newton and other scientists defined the concept of force to be this change in momentum.

My higher secondary book says that momentum[p] is equal to 'mv' where 'm' is mass and 'v' is velocity.Then it was also mentioned that the force[f] was dependent on the rate of change of momentum{[delta]p= m*[delta]v}. Though the book mentions various examples to support that momentum plays a major role in motion, it did not give examples to support that the force depends on the rate of change of momentum{ie.. force also depends on time in addition to momentum}. Could you please give me some experiences from real life to derive the expression for force?

I'm a little unclear as to whether you want a real world phenomenon that exhibits this property or you want the mathematical derivation of the relationship.

For the former, consider yourself landing after a fall. Just before you reach the ground you have some velocity V and so a momentum mV. Soon after that your momentum is brought to zero by the force of the ground pushing up on you. If you stop quickly (say by keeping you legs straight), the ground pushes on you with a much larger force than if you bend your legs at the knees. With the bending, a smaller force acts to change your momentum over a greater time. So the size of the force is a function of how long the force acts.

Mathematically, the rate of change of momentum is
∆(mv)/∆t
=m∆v/∆t
=ma
=F

In language, force is the rate of change of momentum. (If you've studies calculus, you'd say force is the derivative of momentum with respect to time.)

In order to change the momentum of an object the force should be applied to it .The amount of force to be applied depends .It can be
-- small force for long time.
-- big force for shorter time.

The amount of change in the momentum is proportional to the product

Force * time = impulse = change in momentum.

large change in impuse causes big change in momentum.

ex :Suppose a car is motionless due to run out of gas. In order to bring it into motion .If single person pushes the car it takes longer to bring it into motion. i.e small force for long time.
If more people joins applying big force it comes to motion in short time. i.e big force for short time.