Question

A body is moving unidirectionally under the influence of a source of constant power. Its displacement in time t is proportional to ______.

Options

• `t^(1/2)`

• t

• `t^(3/2)`

• `t^2`

Answer 1

A body is moving unidirectionally under the influence of a source of constant power. Its displacement in time t is proportional to `bbunderline(t^(3/2))`.

Explanation:

Power P = Fν is constant.

⇒ `"P" = "m"  "a"  "ν " = "m""dν"/"dt" . "ν "             (∵"a" = "dν"/"dt")`

⇒ `"ν ""dν"/"dt" = "P"/"m" `

⇒ `"ν ""d"  "ν" = "P"/"m""d"  "t"`

On integration, `"ν"^2/2 = "Pt"/"m" + "c"_1`

Let t = 0 at ν = 0 then c₁ = 0

∴ `"ν"^2 = (2"P")/"m" "t"`  or`"ds"/"dt" = sqrt((2"P")/"M")  "t"` (∵) (`"ν" = "ds"/"dt"`)

∴ s = `sqrt((2"P")/"m") ∫ "t"^(1/2) "d" "t"`

s = `sqrt((2"P")/"m") . ("t"^(3/2))/(3/2) + "c"^2`

Let when t = 0 then s = 0 then c₂ = 0

∴ `"s" = 2/3 sqrt((2"P")/"m") "t"^(3/2) => "s" ∝ "t"^(3/2)`