Water flows through a pipe of area 10 cm^2 at 3 m/s. If the pipe narrows to 5 cm^2, the velocity becomes: <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 380 130" style="max-width:380px;margin:0.5em auto;display:block;"> <title>Pipe narrows — continuity equation</title> <rect width="380" height="130" fill="#faf9f5" rx="8"/> <defs><marker id="ah-q09q26" markerWidth="8" markerHeight="6" refX="8" refY="3" orient="auto"><polygon points="0 0, 8 3, 0 6" fill="#788c5d"/></marker></defs> <!-- Wide section --> <path d="M 30,35 L 150,35 L 165,50 L 230,50 L 245,35 L 350,35" fill="none" stroke="#14140f" stroke-width="2"/> <path d="M 30,95 L 150,95 L 165,80 L 230,80 L 245,95 L 350,95" fill="none" stroke="#14140f" stroke-width="2"/> <path d="M 30,35 L 150,35 L 165,50 L 230,50 L 245,35 L 350,35 L 350,95 L 245,95 L 230,80 L 165,80 L 150,95 L 30,95 Z" fill="#788c5d" fill-opacity="0.08"/> <line x1="60" y1="65" x2="110" y2="65" stroke="#788c5d" stroke-width="2" marker-end="url(#ah-q09q26)"/> <text x="70" y="58" font-family="sans-serif" font-size="10" fill="#14140f">v1=3</text> <text x="70" y="115" font-family="sans-serif" font-size="10" fill="#14140f">A1=10 cm^2</text> <line x1="180" y1="65" x2="220" y2="65" stroke="#c6613f" stroke-width="2" marker-end="url(#ah-q09q26)"/> <text x="186" y="58" font-family="sans-serif" font-size="10" fill="#c6613f">v2=?</text> <text x="175" y="115" font-family="sans-serif" font-size="10" fill="#c6613f">A2=5 cm^2</text> <line x1="280" y1="65" x2="330" y2="65" stroke="#788c5d" stroke-width="2" marker-end="url(#ah-q09q26)"/> </svg>

Bernoulli's equation is based on conservation of:

The speed of efflux from a hole at depth 5 m in a large tank is (g = 10 m/s^2):

In a horizontal pipe, if velocity increases at a constriction, the pressure:

Water is flowing through two horizontal pipes of same length connected in series. Their radii are r and 2r. The ratio of pressure drops across them is: <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 400 120" style="max-width:400px;width:100%;height:auto;background:#faf9f5;border-radius:12px;border:1px solid rgba(20,20,19,0.10)"> <title>Two pipes in series — pressure drop proportional to 1/$r^{4}$</title> <defs><marker id="ah-q041-09" markerWidth="8" markerHeight="6" refX="8" refY="3" orient="auto"><polygon points="0,0 8,3 0,6" fill="#788c5d"/></marker></defs> <!-- Narrow pipe (radius r) --> <rect x="40" y="42" width="140" height="16" rx="4" fill="#c6613f" fill-opacity="0.08" stroke="#14140f" stroke-width="1.5"/> <text x="110" y="38" text-anchor="middle" font-size="9" fill="#14140f" font-family="sans-serif">radius r</text> <!-- Wide pipe (radius 2r) --> <rect x="180" y="35" width="140" height="30" rx="6" fill="#788c5d" fill-opacity="0.08" stroke="#14140f" stroke-width="1.5"/> <text x="250" y="32" text-anchor="middle" font-size="9" fill="#14140f" font-family="sans-serif">radius 2r</text> <!-- Flow arrow --> <line x1="60" y1="50" x2="100" y2="50" stroke="#788c5d" stroke-width="1.5" marker-end="url(#ah-q041-09)"/> <line x1="200" y1="50" x2="240" y2="50" stroke="#788c5d" stroke-width="1.5" marker-end="url(#ah-q041-09)"/> <!-- Pressure drop labels --> <text x="110" y="72" text-anchor="middle" font-size="10" font-weight="700" fill="#c6613f" font-family="sans-serif">$\Delta$$P_{1}$ ∝ 1/$r^{4}$</text> <text x="250" y="78" text-anchor="middle" font-size="10" font-weight="700" fill="#788c5d" font-family="sans-serif">$\Delta$$P_{2}$ ∝ 1/(2r)^{4}</text> <!-- Result --> <text x="200" y="105" text-anchor="middle" font-size="12" font-weight="700" fill="#14140f" font-family="sans-serif">$\Delta$$P_{1}$ : $\Delta$$P_{2}$ = 16 : 1</text> </svg>