Suppose that an object of mass {{MATH_103}} is moving along a straight line. If {{MATH_104}} and {{MATH_105}} are the the starting and ending positions, {{MATH_106}} and {{MATH_107}} are the the starting and ending velocities, and {{MATH_108}} is the force acting on the object for any given position, then {{MATH_109}} {{CMD_110}} First we need to get all of our symbolism out in the open. Let: {{CMD_111}} {{CMD_112}} {{MATH_113}} represent position with respect to time, {{CMD_114}} {{MATH_115}} represent velocity with respect to time, {{CMD_116}} {{MATH_117}} represent acceleration with respect to position, {{CMD_118}} {{MATH_119}} represent the starting time, {{CMD_120}} {{MATH_121}} represent the ending time, {{CMD_122}} then we also have that {{CMD_123}} {{CMD_124}} {{MATH_125}} represents the starting position, {{MATH_126}}, {{CMD_127}} {{MATH_128}} represents the ending position, {{MATH_129}}, {{CMD_130}} {{MATH_131}} represents the starting velocity, {{MATH_132}}, {{CMD_133}} {{MATH_134}} represents the ending velocity, {{MATH_135}}. {{CMD_136}} Now write with me, {{MATH_137}} here we are working with functions of distance. We will use the substitution formula, {{CMD_138}} {{ENV_139}} {{CMD_140}} transforming from right to left, to see that {{MATH_141}} and we are now working with functions of time. Since {{MATH_142}}, we may write {{MATH_143}} and now remember that {{MATH_144}}, so {{MATH_145}} However, {{MATH_146}}, so rearranging we have, {{MATH_147}} Now we apply the substitution formula again, this time we will transform left to right {{CMD_148}} {{ENV_149}} {{CMD_150}} and so we see {{MATH_151}} and we are working with functions of velocity. At last, setting {{MATH_152}} and {{MATH_153}}, we can evaluate this integral, {{ENV_154}} {{CMD_155}} {{CMD_156}}