A particle of unit mass undergoes one-dimensional motion such that its velocity varies according to v ( x ) = x - 2 n , where and n are constants and x is the position of the particle. The acceleration of the particle as a function of x , is given by

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Q.
A particle of unit mass undergoes one-dimensional motion such that its velocity varies according to $v (x) = β x^{- 2 n},$ where $β$ and $n$ are constants and $x$ is the position of the particle. The acceleration of the particle as a function of $x$ , is given by [2015]

1 $- 2 β^{2} x^{- 2 n + 1}$

2 $- 2 n β^{2} e^{- 4 n + 1}$

3 $- 2 n β^{2} x^{- 2 n - 1}$

4 $- 2 n β^{2} x^{- 4 n - 1}$

Ans.
(4)

According to question, velocity of unit mass varies as

$v (x) = β x^{- 2 n}$ ...(i), $\frac{d v}{d x} = - 2 n β x^{- 2 n - 1}$ ...(ii)

Acceleration of the particle is given by

$a = \frac{d v}{d t} = \frac{d v}{d x} \times \frac{d x}{d t} = \frac{d v}{d x} \times v$

Using equation (i) and (ii), we get

$a = (- 2 n β x^{- 2 n - 1}) \times (β x^{- 2 n}) = - 2 n β^{2} x^{- 4 n - 1}$

Solution

Q.
A particle of unit mass undergoes one-dimensional motion such that its velocity varies according to $v (x) = β x^{- 2 n},$ where $β$ and $n$ are constants and $x$ is the position of the particle. The acceleration of the particle as a function of $x$ , is given by [2015]

1 $- 2 β^{2} x^{- 2 n + 1}$

2 $- 2 n β^{2} e^{- 4 n + 1}$

3 $- 2 n β^{2} x^{- 2 n - 1}$

4 $- 2 n β^{2} x^{- 4 n - 1}$

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Solution

Q. A particle of unit mass undergoes one-dimensional motion such that its velocity varies according to v(x)=βx-2n, where β and n are constants and x is the position of the particle. The acceleration of the particle as a function of x, is given by [2015]

1 -2β2x-2n+1

2 -2nβ2e-4n+1

3 -2nβ2x-2n-1

4 -2nβ2x-4n-1

Ans. (4) According to question, velocity of unit mass varies as v(x)=βx-2n ...(i), dvdx=-2nβx-2n-1 ...(ii) Acceleration of the particle is given by a=dvdt=dvdx×dxdt=dvdx×v Using equation (i) and (ii), we get a=(-2nβx-2n-1)×(βx-2n)=-2nβ2x-4n-1

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Q.
A particle of unit mass undergoes one-dimensional motion such that its velocity varies according to $v (x) = β x^{- 2 n},$ where $β$ and $n$ are constants and $x$ is the position of the particle. The acceleration of the particle as a function of $x$ , is given by [2015]

1 $- 2 β^{2} x^{- 2 n + 1}$

2 $- 2 n β^{2} e^{- 4 n + 1}$

3 $- 2 n β^{2} x^{- 2 n - 1}$

4 $- 2 n β^{2} x^{- 4 n - 1}$