Impulse theorem

Question

According to my physics book, The impulse-momentum theorem states that the change in momentum of an object equals the impulse applied to it.

To indicate the average force acting on the object itself, the book uses the following notation $<vec F>$.

I would like to understand how to indicate the magnitude of this average force. Shall I use the notation $|<vec F>|$ or $<|vec F|>$?
I am not sure the second notation is correct.

forces newtonian mechanics

AFG · Accepted Answer

The first one is the magnitude of the average force $$|<vec{F}>|=Bigg|frac{1}{T}int_0^Tvec{F}dtBigg|.$$

The second one is the average of the magnitude of the force,

$$<|vec{F}|>=frac{1}{T}int_0^T|vec{F}|dt.$$

In the impulse-momentum context, the first one is which makes sense.

Bill N · Answer

Like many other mathematical operations in physics, the order can be important. For example, the integral of squares is not the same as the square of an integral:
$$int_0^1 x^2 mathrm d x ne left( int_0^1 x mathrm d xright)^2$$.
In your case, try the following example:

Average the numbers 2, -3, 4, 6, -1.
Average their absolute values.

Are they the same? That should tell you what you want to know.
Or consider a force, $vec{F}(t)=2.0 sin(pi t)hat{i} $ acting for 2.0 seconds. Then try the absolute value of that acting for 2.0 seconds.

Impulse theorem

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