Trying to prove that Impulse = change in momentum

[GRB 080319B]

Homework Statement

I was trying to prove that Impulse = change in momentum. I end up proving it, but I'm not sure if the two assumptions I made are true. I also am not sure if my initial equations are completely true (I got them from a conceptual physics book which isn't heavy on equations themselves: I'm not sure if the equations they are using are instantaneous or average). I apologize for the notation; I don't know how to use latex. Thank you.

Homework Equations

I=Ft
F=ma
a=(delta v/ delta t)
p=mv

The Attempt at a Solution

I=Ft
I=(ma)t
I=m(delta v/ delta t)t
I=m(delta v) Here I made an assumption that the time in the initial impulse function was equivalent to the delta time for the velocity in order to cancel the two to get velocity by itself.
I=m(v1 - v0) The 1s and 0s signify final and initial values respectively.
I=m((p1/m1) - (p0/m0))
I=m(1/m)(p1 - p0) Here I made an assumption that the initial and final masses were equivalent to each other and to the mass from the 2nd law substitution I performed earlier in order to cancel the two to get the momentum by itself.
I=(p1 - p0)
I= delta p

 

[kmikias]

you should include mass into your derivative because somethimes mass makes the different so J = d(mv)/dt ...so mv=p ...then J = d(p)/dt...

 

[GRB 080319B]

What notation do you mean by J, do you mean joules? Also, do these equations involve derivatives or just difference quotients? My first assumption was based on treating the acceleration as delta v / delta t, not dv/dt. However, you seem to be using the derivative, which leaves me in a quandary: I want to end up with delta p instead of a derivative, but can't seem to do so. Is acceleration treated as instantaneous, and are the other parts right?

 

[heth]

What notation do you mean by J, do you mean joules?

J is sometimes used to represent impulse.

 

[GRB 080319B]

On wikipedia, gives the equation as I = F(delta t) = m(delta v) = delta p , assuming that the mass is constant. That's where become confused, because I'm not sure if I can assume that the mass doesn't change according to my work above. Should the mass from the 2nd law and the mass of the momentum be equivalent?

 

[heth]

The 'universal' definition that you can use for billiard balls going at everyday speeds, and also for something like a rocket (mass changes as fuel burns) and also for photons (which have zero mass but do have momentum) and also for relativity, is:

Impulse = Fdt = dp

(since F = dp/dt)

Strictly speaking, this is the 'correct' form of Newton's 2nd Law, F = ma is what you get if the mass is constant.

As you say, if you're dealing with everyday objects, you can assume that the mass is constant.

It's not clear from the problem statement which equations you're supposed to start with to "prove" this.