Editing Impulse (physics) (section)

==Mathematical derivation in the case of an object of constant mass==
[[File:Happy vs. Sad Ball.webm|thumbnail|The impulse delivered by the "sad" ball is {{math|''mv''<sub>0</sub>}}, where {{math|''v''<sub>0</sub>}} is the speed upon impact.  To the extent that it bounces back with speed {{math|''v''<sub>0</sub>}}, the "happy" ball delivers an impulse of {{math|1=''m''&Delta;''v'' = 2''mv''<sub>0</sub>}}.<ref>[http://materialseducation.org/educators/matedu-modules/docs/Property_Differences_in_Polymers.pdf Property Differences In Polymers: Happy/Sad Balls]</ref>]]
Impulse {{math|'''J'''}} produced from time {{math|''t''<sub>1</sub>}} to {{math|''t''<sub>2</sub>}} is defined to be{{sfn|Serway|Jewett|2004|loc=chpt. 9.2 Impulse and Momentum}}
<math display=block qid=Q837940>\mathbf{J} = \int_{t_1}^{t_2} \mathbf{F}\, \mathrm{d}t,</math>
where {{math|'''F'''}} is the resultant force applied from {{math|''t''<sub>1</sub>}} to {{math|''t''<sub>2</sub>}}.

From [[Newton's laws of motion#Newton's second law|Newton's second law]], force is related to [[momentum]] {{math|'''p'''}} by
<math display=block>\mathbf{F} = \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t}.</math>

Therefore,
<math display=block qid=Q837940>\begin{align}
 \mathbf{J} &= \int_{t_1}^{t_2} \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t}\, \mathrm{d}t \\
 &= \int_{\mathbf{p}_1}^{\mathbf{p}_2} \mathrm{d}\mathbf{p} \\
 &= \mathbf{p}_2 - \mathbf{p} _1= \Delta \mathbf{p}, \end{align}</math>
where {{math|Δ'''p'''}} is the change in linear momentum from time {{math|''t''<sub>1</sub>}} to {{math|''t''<sub>2</sub>}}. This is often called the impulse-momentum theorem (analogous to the [[work-energy theorem]]).

As a result, an impulse may also be regarded as the change in momentum of an object to which a resultant force is applied. The impulse may be expressed in a simpler form when the mass is constant:
<math display=block qid=Q837940>\mathbf{J} = \int_{t_1}^{t_2} \mathbf{F}\, \mathrm{d}t =  \Delta\mathbf{p} = m \mathbf{v_2} - m \mathbf{v_1},</math>

where
*{{math|'''F'''}} is the resultant force applied,
*{{math|''t''<sub>1</sub>}} and {{math|''t''<sub>2</sub>}} are times when the impulse begins and ends, respectively,
*{{mvar|m}} is the mass of the object,
*{{math|'''v'''<sub>2</sub>}} is the final velocity of the object at the end of the time interval, and
*{{math|'''v'''<sub>1</sub>}} is the initial velocity of the object when the time interval begins.

Impulse has the same units and dimensions {{nowrap|(MLT<sup>&minus;1</sup>)}} as momentum. In the [[International System of Units]], these are {{nowrap|1=[[kilogram|kg]]⋅[[meter per second|m/s]] =}} {{nowrap|[[newton (units)|N]]⋅[[second|s]]}}. In [[English engineering units]], they are {{nowrap|1=[[Slug (unit)|slug]]⋅[[foot per second|ft/s]] =}} {{nowrap|[[pound (force)|lbf]]⋅[[second|s]]}}.

The term "impulse" is also used to refer to a fast-acting force or [[Impact (mechanics)|impact]]. 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 function|step change]], and is not physically possible. However, this is a useful model for computing the effects of ideal collisions (such as in videogame [[physics engine]]s).  Additionally, in rocketry, the term "total impulse" is commonly used and is considered synonymous with the term "impulse".