Final answer:
Using conservation of energy principles, the maximum height for water exiting a garden hose with a nozzle is approximately 27.6 meters, and without the nozzle is roughly 0.187 meters. The significantly higher exit velocity through the nozzle results in the water reaching a much greater height.
Step-by-step explanation:
The maximum height to which water could be squirted using a garden hose with and without a nozzle refers to how high the water can reach solely under the influence of the velocity with which it exits the hose. We can use the principles of conservation of energy and equations of motion for this calculation. With the nozzle, the water exits at a higher velocity due to the constriction, whereas without the nozzle it exits at a lower velocity.
Calculate the Maximum Height with Nozzle
When the water exits the nozzle at a velocity of 23.2 m/s, we assume that the kinetic energy is completely converted to potential energy at the maximum height. The equation used is:
Kinetic Energy = Potential Energy
0.5 × m × v^2 = m × g × h
Solving for h, we have:
h = (v^2) / (2 × g)
h = (23.2^2 m^2/s^2) / (2 × 9.81 m/s^2)
h ≈ 27.6 meters
Calculate the Maximum Height with Nozzle Removed
Similarly, if the nozzle is removed and the water exits the hose at 1.92 m/s, the maximum height is calculated with the same energy conversion principle:
h = (v^2) / (2 × g)
h = (1.92^2 m^2/s^2) / (2 × 9.81 m/s^2)
h ≈ 0.187 meters
We can see that the water reaches a significantly higher maximum height when exiting through the nozzle due to its higher exit velocity.