(III) Suppose the opening in the tank of Fig. 10-54 is a...

(III) Suppose the opening in the tank of Fig. $10-54$ is a height $h_{1}$ above the base and the liquid surface is a height $h_{2}$ above the base. The tank rests on level ground. (a) At what horizontal distance from the base of the tank will the fluid strike the ground? (b) At what other height, $h_{1}^{\prime}$ can a hole be placed so that the emerging liquid will have the same "range"? Assume $v_{2} \approx 0$.

Key Concepts

Optimization of Exit Point

Determining the optimal or equivalent positions for holes in a fluid container often requires balancing the effects of exit velocity and the vertical distance through which the fluid falls. A hole placed lower on the tank benefits from a higher exit speed due to greater pressure, but a higher hole increases the flight time. Understanding this trade-off is essential when analyzing and optimizing the horizontal range of the fluid jet.

Energy Conservation in Fluids

The principle of energy conservation underlies the derivation of the fluid's exit velocity. As the fluid drops from a higher potential, its potential energy is converted into kinetic energy, which is then manifested in the velocity at the orifice. This concept is instrumental in linking gravitational effects to fluid flow dynamics.

Projectile Motion

Once the fluid exits the tank through the hole, it follows a parabolic trajectory under the influence of gravity. Projectile motion analysis involves decomposing the movement into horizontal and vertical components to determine the range and time of flight. This concept is crucial when calculating where the fluid will land on level ground.

Torricelli’s Law

This law relates the speed of fluid flowing out of an orifice to the height difference between the fluid surface and the orifice, based on the conversion of gravitational potential energy into kinetic energy. It is fundamental in determining the exit velocity of the fluid, which is central to solving problems involving fluid discharge from containers.

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