# Mass = m, momentum is p=mv. In time Δt, momentum changes by Δp, the rate of change of momentum is: Δp/Δt=Δ(mv)/t=mΔv/Δt

Mass = m , momentum is $p=mv$. In time $\mathrm{\Delta }t$, momentum changes by $\mathrm{\Delta }p$, the rate of change of momentum is:
$\frac{\mathrm{\Delta }p}{\mathrm{\Delta }t}=\frac{\mathrm{\Delta }\left(mv\right)}{t}=m\frac{\mathrm{\Delta }v}{\mathrm{\Delta }t}$
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ralharn
1) Yes indeed, the absence of a $\mathrm{\Delta }$ in the second expression is just a typo.
2) The last expression is derived assuming that mass is a constant. If it helps, just set the mass equal to 4, or something. If we want to know how the quantity 4v changes, we really only need to know how the quantity v changes. Suppose v changes from ${v}_{1}$ to ${v}_{2}$. Then
$\mathrm{\Delta }v={v}_{2}-{v}_{1}\phantom{\rule{thinmathspace}{0ex}},$
and what will $\mathrm{\Delta }\left(4v\right)$ be? Why it will be
$\mathrm{\Delta }\left(4v\right)=4{v}_{2}-4{v}_{1}=4\left({v}_{2}-{v}_{1}\right)=4\mathrm{\Delta }v\phantom{\rule{thinmathspace}{0ex}}.$
So the constant comes out the front, because it doesn't change between the start and end points that you're considering. Hence it can be factored out.
3) It's important that I point out that
$\frac{\mathrm{\Delta }p}{\mathrm{\Delta }t}=m\frac{\mathrm{\Delta }v}{\mathrm{\Delta }t}$
is not Newton's second law. It's just a relationship between the rate of change of momentum and acceleration, and can be derived straight from the definitions of these things. Newton's second law cannot be derived, and is a statement of real physical content --- hence it is called a law. Newton's law can be written as either
$F=m\frac{\mathrm{\Delta }v}{\mathrm{\Delta }t}\phantom{\rule{thinmathspace}{0ex}},$
or
$F=\frac{\mathrm{\Delta }p}{\mathrm{\Delta }t}\phantom{\rule{thinmathspace}{0ex}},$
whichever you prefer. You can show these two are equivalent using the argument in your textbook (although in fact they're not quite equivalent, as I established above --- the first equation only holds when mass is constant; the second is more general, and more universally true).