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

Answer 1

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")`