Derivation of expression for force

[Alpharup]

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?

 

[timthereaper]

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.

 

[Fewmet]

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.)

 

[srivi]

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.

 

[PJC01]

The expression for force is derived from Newton's Second Law of Motion, which states that the force applied to an object is directly proportional to the rate of change of its momentum. This can be mathematically expressed as F = dp/dt, where F is the force, dp is the change in momentum, and dt is the change in time.

To better understand this concept, let's look at some real-life examples. Imagine you are pushing a shopping cart in a straight line. The force you apply to the cart is dependent on how quickly you are able to change its momentum. If you push the cart with a constant force, it will accelerate at a constant rate. However, if you push the cart with a larger force, it will accelerate at a faster rate, since the change in momentum will be greater.

Another example is a car accelerating on a straight road. The engine of the car applies a force to the wheels, which in turn causes the car to accelerate. The rate of change of the car's momentum is directly related to the force applied by the engine. If the engine produces a greater force, the car will accelerate faster.

In both of these examples, we can see that the force applied is directly proportional to the rate of change of momentum. This supports the expression F = dp/dt and demonstrates how force is dependent on both momentum and time.

In conclusion, the expression for force is derived from real-life experiences and observations, and it is supported by Newton's Second Law of Motion. By understanding how force is related to the rate of change of momentum, we can better understand the fundamental principles of motion and how objects behave in the physical world.