Definitions

# impulse

[im-puhls]
impulse, in mechanics: see momentum.

In classical mechanics, an impulse is defined as the integral of a force with respect to time:

$mathbf\left\{I\right\} = int mathbf\left\{F\right\}, dt$
where
I is impulse (sometimes marked J),
F is the force, and
dt is an infinitesimal amount of time.

A simple derivation using Newton's second law yields:

$mathbf\left\{I\right\} = int frac\left\{dmathbf\left\{p\right\}\right\}\left\{dt\right\}, dt$
$mathbf\left\{I\right\} = int dmathbf\left\{p\right\}$
$mathbf\left\{I\right\} = Delta mathbf\left\{p\right\}$
where
p is momentum

This is often called the impulse-momentum theorem.

As a result, an impulse may also be regarded as the change in momentum of an object to which a force is applied. The impulse may be expressed in a simpler form when both the force and the mass are constant:

$mathbf\left\{I\right\} = mathbf\left\{F\right\}Delta t = m Delta mathbf\left\{v\right\} = Delta p$

where

F is the constant total net force applied,
$Delta t$ is the time interval over which the force is applied,
m is the constant mass of the object,
Δv is the change in velocity produced by the force in the considered time interval, and
v = Δ(mv) is the change in linear momentum.

However, it is often the case that one or both of these two quantities vary.

In the technical sense, impulse is a physical quantity, not an event or force. However, the term "impulse" is also used to refer to a fast-acting force. This type of impulse is often idealized so that the change in momentum produced by the force happens with no change in time. This sort of change is a step change, and is not physically possible. However, this is a useful model for certain purposes, such as computing the effects of ideal collisions, especially in game physics engines.

Impulse has the same units and dimensions as momentum (kg m/s = N·s).

Using basic math, Impulse can be calculated using the equation:

$mathbf\left\{F\right\}t = Delta p$

$Delta p$ can be calculated, if initial and final velocities are known, by using "mv(f) - mv(i)" or otherwise known as "mv - mu"

where

F is the constant total net force applied,
$t$ is the time interval over which the force is applied,
m is the constant mass of the object,
v is the final velocity of the object at the end of the time interval, and
u is the initial velocity of the object when the time interval begins.

Hence: $mathbf\left\{F\right\}t = mv - mu$