Q.Water flows steadily through a pipe that rises from a lower point, where its speed is $2.0\ \text{m/s}$ and the pressure is $2.0\times10^{5}\ \text{Pa}$, to a higher point $5.0\ \text{m}$ above it, where the pipe has narrowed so that the speed has increased to $4.0\ \text{m/s}$. Using Bernoulli's theorem, calculate the pressure of the water at the higher point. (Density of water $= 1000\ \text{kg/m}^3$; take $g = 9.8\ \text{m/s}^2$.)

Given: $P_1 = 2.0\times10^5\ \text{Pa}$, $v_1 = 2.0\ \text{m/s}$, $h_1 = 0$; $v_2 = 4.0\ \text{m/s}$, $h_2 = 5.0\ \text{m}$; $\rho = 1000\ \text{kg/m}^3$, $g = 9.8\ \text{m/s}^2$.

By Bernoulli's theorem, $P_1 + \tfrac12\rho v_1^2 + \rho g h_1 = P_2 + \tfrac12\rho v_2^2 + \rho g h_2$, so:

$P_2 = 2.0\times10^5 + 500\times(-12) + 9800\times(-5.0) = 2.0\times10^5 - 6000 - 49{,}000$ …

Write Bernoulli's theorem between the two points and solve for the unknown pressure $P_2$, carefully substituting the correc …

• Forgetting to include the gravitational potential-energy term $\rho g(h_1-h_2)$ when the pipe changes height, and only accounting for the change in speed. …