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## Impulse & Momentum Discussion

Momentum keeps me going.

From the *Principia*.

Quantitas motus est mensura ejusdem orta ex velocitate et quantite materiæ conjunctim.The quantity of motion is the measure of the same, arising from the velocity and the quantity of matter conjunctly.

We now call this quantity momentum. Momentum is resistance to stopping. It’s a kind of inertia for moving bodies.

**p** = *m***v**

Units

[kg m/s]

More form the *Principia*.

Mutationem motus proportionalem esse vi motrici impressæ, & fieri secundum lineaum rectam qua vis illa imprimitur. The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

Force is directly proportional to the rate of change of momentum with time.

**∑F = ma**

**F = m Δv/Δt**

**F̅Δ t = mΔv**

The quantity on the right is the change in momentum (Δ**p** = *m*Δ**v**). That should be easily seen. The quantity on the right is something new. We’ll call it impulse and represent it with the letter **J**. (I’ve also seen the more sensible letter **I** used from time to time.) Thus…

**J** = **F̅**Δ*t*

or in the language of calculus

J = | ⌠⌡ | F dt |

Units

[N s]

This relationship is called the impulse-momentum theorem. In words “impulse causes a change in momentum”.

**J** = Δ**p**

Maybe because the use of the letter “J” to represent a quantity whose name begins with the letter “I” is so odd, this relationship is usually written in its expanded form…

**F̅**Δ*t* = *m*Δ**v**

or in it’s calculus form…

⌠ ⌡ | F dt = Δp |

In a way, this is a nice convention since now we can see the equivalence of units a bit more easily.

[N s = kg m/s]