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Question
If F is the constant force generated by the motor of an automobile of mass M, its velocity V is given by `"M""dv"/"dt"` = F – kV, where k is a constant. Express V in terms of t given that V = 0 when t = 0
Solution
Given equation `"M" "dV"/"dt"` = F – kV ......(∵ F and k are constant)
`"M" "dV"/"dt" = "k"("F"/"k" - "V")`
`int "dV"/(("F"/"k" - "V")) = "k"/"M" int "dt"`
`- log ("F"/"k" - "V") = "k"/"M" "t" + "c"` .......(1)
Given V = 0 and t = 0
⇒ `- log "F"/"k"` = c
Substituting in (1)
`- log ("F"/"k" - "V") = "kt"/"M" - log ("F"/"k")`
`log("F"/"k") - log (("F"- "Vk")/"k") = "kt"/"M"`
`log (("F"/"k")/(("F" - "Vk")/"k")) = "kt"/"M"`
`("F"/("F" - "Vk")) = "e"^("kt"/"M")`
F = `("F" - "Vk") "e"^("kt"/"M")`
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