/ s ). ▁The ▁corresponding ▁English ▁engineering ▁units ▁are ▁the ▁p ound - second ▁( l bf ⋅ s ) ▁and ▁the ▁sl ug - foot ▁per ▁second ▁( slug ⋅ ft / s ). ▁ ▁A ▁result ant ▁force ▁causes ▁acceleration ▁and ▁a ▁change ▁in ▁the ▁velocity ▁of ▁the ▁body ▁for ▁as ▁long ▁as ▁it ▁acts . ▁A ▁result ant ▁force ▁applied ▁over ▁a ▁longer ▁time ▁therefore ▁produces ▁a ▁bigger ▁change ▁in ▁linear ▁momentum ▁than ▁the ▁same ▁force ▁applied ▁briefly : ▁the ▁change ▁in ▁momentum ▁is ▁equal ▁to ▁the ▁product ▁of ▁the ▁average ▁force ▁and ▁duration . ▁Con vers ely , ▁a ▁small ▁force ▁applied ▁for ▁a ▁long ▁time ▁produces ▁the ▁same ▁change ▁in ▁momentum — the ▁same ▁imp ulse — as ▁a ▁larger ▁force ▁applied ▁briefly . ▁ ▁The ▁imp ulse ▁is ▁the ▁integral ▁of ▁the ▁result ant ▁force ▁( F ) ▁with ▁respect ▁to ▁time : ▁ ▁Mathemat ical ▁deriv ation ▁in ▁the ▁case ▁of ▁an ▁object ▁of ▁constant ▁mass ▁ ▁Imp ulse ▁J ▁produced ▁from ▁time ▁t 1 ▁to ▁t 2 ▁is ▁defined ▁to ▁be ▁ ▁where ▁F ▁is ▁the ▁result ant ▁force ▁applied ▁from ▁t 1 ▁to ▁t 2 . ▁ ▁From ▁Newton ' s ▁second ▁law , ▁force ▁is ▁related ▁to ▁momentum ▁p ▁by ▁ ▁Therefore , ▁ ▁where ▁ Δ p ▁is ▁the ▁change ▁in ▁linear ▁momentum ▁from ▁time ▁t 1 ▁to ▁t 2 . ▁This ▁is ▁often ▁called ▁the ▁imp ulse - m oment um ▁theorem ▁( anal og ous ▁to ▁the ▁work - energy