91Ó°ÊÓ

The change in blood pressure as a person stands up can be modeled as the change in fluid pressure at different heights. Hence, it is fit to use the Bernoulli equation.

The mathematical form of Bernoulli's equation is as follows

$p_1+\frac{1}{2}\mathrm{ÒÏ}{v}_1^2+\mathrm{ÒÏ}g{y}_1=p_2+\frac{1}{2}\mathrm{ÒÏ}{v}_2^2+\mathrm{ÒÏ}g{y}_2$

Since the speed of the blood is considered as uniform, the above equation can be written as follows.

$p_1+\mathrm{ÒÏ}g{y}_1=p_2+\mathrm{ÒÏ}g{y}_2$

The variables in the above equation are described as follows.

$p_1$: Blood pressure lying down, given as $118mmHg$

$p_2$: Blood pressure standing up, unknown

$y_1$: Height lying down

$y_2$: Height standing up

$\mathrm{ÒÏ}$: Density of blood $1060kg/m^3$

$g$ : Acceleration due to gravity,$9.8m/s^2$

Isolate for $p_2$from above equation.

$p_1+\mathrm{ÒÏ}g{y}_1=p_2+\mathrm{ÒÏ}g{y}_2$

$p_2=p_1+\mathrm{ÒÏ}g\left(y_1-y_2\right)$

$p_2=p_1-\mathrm{ÒÏ}g\left(y_2-y_1\right)$

Substitute$1060kg/m^3$for density of blood, $9.8m/s^2$for $g$, $118mmHg$ for $p_1$and $0.40m$for $y_2-y_1$which is the distance between the heart and the brain, the same amount of displacement blood makes between heart and brain when person stands up

$p_2=118\mathrm{mmHg}-\left(1060\mathrm{kg}/{\mathrm{m}}^3\right)\left(9.8\mathrm{m}/{\mathrm{s}}^2\right)(0.40\mathrm{m})$

$=118\mathrm{mmHg}-4155.2\mathrm{Pa}$

$=118\mathrm{mmHg}-4155.2\mathrm{Pa}\left(\frac{0.0075\mathrm{mmHg}}{1\mathrm{Pa}}\right)$

$=87\mathrm{mmHg}$

This blood pressure ,at $87mmHg$, will cause nausea.