First slide

Instantaneous acceleration

Question

A point moves in a straight line so that its displacement x metre at time t second is given by x² = 1 + t². Its acceleration in ms^-2 at time t second is

Moderate

A

$\frac{1}{x^{3}}$
B

$\frac{- t}{x^{2} t}$
C

$\frac{1}{x} - \frac{t^{2}}{x^{3}}$
D

$\frac{1}{x} - \frac{1}{x^{2}} t$

Solution

$x^{2} = 1 + t^{2} or x = {(1 + t^{2})}^{1 / 2} \frac{dx}{dt} = \frac{1}{2} {(1 + t^{2})}^{- 1 / 2} 2 t = t {(1 + t^{2})}^{- 1 / 2} a = \frac{d^{2} x}{{dt}^{2}} = {(1 + t^{2})}^{- 1 / 2} + t (- \frac{1}{2}) {(1 + t^{2})}^{- 3 / 2} 2 t = \frac{1}{x} - \frac{t^{2}}{x^{3}}$

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