# Is Impulse the anti derivative of momentum?

1. Feb 5, 2016

### BreCheese

Is Impulse an anti derivative of momentum? I know that momentum is an anti derivative of force (proof below), but I'm struggling with understanding the difference between momentum and impulse. My thoughts led me to think that both impulse and momentum are anti derivatives of force, but I'm not certain that impulse is an anti derivative of momentum.
Here is an analogy I've thought of:
If y=position, then y'=velocity, and y''=acceleration. so the anti derivative of acceleration is velocity, and the anti derivative of velocity is position.
So, similarly, I'm thinking that
If y=impulse, then y'=momentum, and y''=force. So the anti derivative of force is momentum, and the anti derivative of momentum is force. (is this right?)
Proof that an anti derivative of force if momentum:
p=mv ; read, "momentum is equal to mass multiplied by velocity"
F=ma ; a=dv/dt
F=m(dv/dt)
F=d/dt(mv) ; p=mv
F=d/dt(p) ; read, "Force is equal to the derivative of "p" (AKA: momentum) with respect to time." or in other words, "The derivative of momentum is force."

Thoughts about momentum being a derivative of impulse:
F=d/dt(p)
F(dt)=dp
FΔt=Δp
F=Δp/Δt ; Force (net) equals the change in momentum divided by the change in time.
Impulse (J) = the integral of Force (net) from time a to time b.
Impulse (J)= the integral of Δp/Δt from time a to time b.
The "Δp/Δt" reminds me of a similar formula for finding slope: y2-y1/x2-x1 or more simply, Δy/Δx.
My question is, is Δp/Δt the slope of the line tangent to some function? Is this aforementioned function momentum, or perhaps average momentum?
If so, then that is like saying Impulse (J)= the integral of momentum(average) from time a to time b. In other words, Impulse is the anti derivative of momentum.
Somehow I confused myself, and don't know if my conclusion is correct. I'm struggling with myself about my logic.
Any help/clarification would be appreciated,
Thanks.

2. Feb 5, 2016

### PeroK

Impulse is the change in momentum.

$J = \int_{t_1}^{t_2} Fdt = \int_{t_1}^{t_2} \frac{dp}{dt} dt = p(t_2) - p(t_1)$