The relation between impulse and momentum

[greg_rack]

I get that the impulse is I=FΔt, but why is it even equal to change in momentum caused by the force under consideration?

 

[etotheipi]

$$\vec{I} = \int_{t_1}^{t_2} \vec{F} dt = \int_{t_1}^{t_2} \frac{d\vec{P}}{dt} dt = \vec{P}(t_2) - \vec{P}(t_1) = \Delta \vec{P}$$$\vec{F}$ is the net force on the particle or extended body.

 

[greg_rack]

$$I = \int_{t_1}^{t_2} \vec{F} \cdot dt = \int_{t_1}^{t_2} \frac{d\vec{P}}{dt} dt = \vec{P}(t_2) - \vec{P}(t_1) = \Delta \vec{P}$$$\vec{F}$ is the net force on the particle or extended body.

I still haven't studied integrals...

 

[etotheipi]

For constant $\vec{F}$,$$\vec{I} = \vec{F} \Delta t = \frac{d\vec{P}}{dt} \Delta t = \Delta \vec{P}$$

 

[greg_rack]

For constant $\vec{F}$,$$I = \vec{F} \Delta t = \frac{d\vec{P}}{dt} \Delta t = \Delta \vec{P}$$

Thank you so much, now it's clear!

 

[Lnewqban]

Summary:: I've studied momentum and impulse a long time ago, don't judge the silliness of this question

I get that the impulse is I=FΔt, but why is it even equal to change in momentum caused by the force under consideration?

Please, see:
https://en.m.wikipedia.org/wiki/Impulse_(physics)

https://en.m.wikipedia.org/wiki/Momentum

"A resultant force causes acceleration and a change in the velocity of the body for as long as it acts. A resultant force applied over a longer time therefore produces a bigger change in linear momentum than the same force applied briefly: the change in momentum is equal to the product of the average force and duration. Conversely, a small force applied for a long time produces the same change in momentum—the same impulse—as a larger force applied briefly."

 

[greg_rack]

Please, see:
https://en.m.wikipedia.org/wiki/Impulse_(physics)

https://en.m.wikipedia.org/wiki/Momentum

"A resultant force causes acceleration and a change in the velocity of the body for as long as it acts. A resultant force applied over a longer time therefore produces a bigger change in linear momentum than the same force applied briefly: the change in momentum is equal to the product of the average force and duration. Conversely, a small force applied for a long time produces the same change in momentum—the same impulse—as a larger force applied briefly."

Really exhaustive explanation, thank you!

 

[Lnewqban]

You are welcome.
These basic concepts are very important.
Note units related to impulse and momentum in each system.