A particle moves in a straight line such that its displacement, $s$ metres, from a fixed point, at time $t$ seconds, $t \geqslant 0$, is given by $s=(1+3 t)^{-\frac{1}{2}}$. (i) Find the exact speed of the particle when $t=1$.

Explanation

To find the exact speed of the particle when t = 1, we first need to find the velocity function. We differentiate the displacement function s = (1 + 3t)^(-1/2) with respect to time t. We get v = d/dt ((1 + 3t)^(-1/2)) = (-1/2) (1 + 3t)^(-3/2) * 3 = -3/2 (1 + 3t)^(-3/2). Then, we substitute t = 1 into the velocity function to get v(1) = -3/2 (1 + 3*1)^(-3/2) = -3/2 (4)^(-3/2) = -3/2 * 1/4 = -3/8. The speed is the absolute value of the velocity, so the exact speed of the particle when t = 1 is 3/8.

A particle moves in a straight line such that its displacement, $s$ metres, from a fixed point, at time $t$ seconds, $t \geqslant 0$, is given by $s=(1+3 t)^{-\frac{1}{2}}$. (i) Find the exact speed of the particle when $t=1$.

Explanation

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