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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`
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Solution
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)`
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