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The displacement of a particle is given by $s = 3t^3 + 7t^2 + 5t + 8$ metres, where $t$ is in seconds. The acceleration at $t = 1\text{ s}$ is:
Velocity: $v = \dfrac{ds}{dt} = 9t^2 + 14t + 5$
Acceleration: $a = \dfrac{dv}{dt} = 18t + 14$
At $t = 1\text{ s}$: $a = 18(1) + 14 = \mathbf{32\text{ m/s}^2}$
Gastric juice secretion is stimulated by which hormone?
Gastrin, released by G cells in the stomach antrum, stimulates parietal cells to secrete HCl and chief cells to secrete pepsinogen. Cholecystokinin and secretin are involved in intestinal digestion; insulin regulates blood glucose.
A particle is projected vertically upward with $u = 10\text{ m/s}$. Air exerts a resistive force $F = -0.2v^2$ on it ($m = 2\text{ kg}$, $g = 10\text{ m/s}^2$). The maximum height attained is:
Net force (taking up as positive): $F_{net} = -mg - 0.2v^2 = -20 - 0.2v^2$
Using $ma = v\dfrac{dv}{dx}$: $2v\dfrac{dv}{dx} = -20 - 0.2v^2$
Separating variables: $\dfrac{v\,dv}{20+0.2v^2} = -\dfrac{dx}{2}$
Integrating from $v = 10$ to $v = 0$: let $k = 20 + 0.2v^2$, $dk = 0.4v\,dv$
$\dfrac{1}{0.4}\big[\ln(20) - \ln(40)\big] = -\dfrac{H}{2} \Rightarrow H = \dfrac{2\ln 2}{0.4} = \mathbf{5\ln 2}$
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