First slide

Instantaneous acceleration

Question

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

Difficult

A

$- 2 β^{2} x^{- 2 n + 1}$
B

$- 2 n β^{2} e^{- 4 n + 1}$
C

$- 2 n β^{2} x^{- 2 n - 1}$
D

$- 2 n β^{2} x^{- 4 n - 1}$

Solution

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

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