You are watering your lawn with a hose when you put your finger over the hose opening to increase the distance the water reaches. If you are holding the hose horizontally, and the distance the water reaches increases by a factor of 4, what fraction of the hose opening did you block?

검증된 단계별 안내

Understand the relationship between the range of the water stream and the velocity of the water exiting the hose. The horizontal range of a projectile (in this case, the water) is proportional to the square of its initial velocity: \( R \propto v^2 \). Since the range increases by a factor of 4, the velocity must increase by a factor of \( \sqrt{4} = 2 \).

Recognize that the velocity of the water exiting the hose is related to the pressure and the cross-sectional area of the hose opening. According to the principle of conservation of mass (continuity equation), the flow rate \( Q \) is constant: \( Q = A v \), where \( A \) is the cross-sectional area of the opening and \( v \) is the velocity of the water.

Since the velocity increases by a factor of 2, the cross-sectional area of the opening must decrease to maintain the same flow rate. Specifically, \( A_{new} = \frac{A_{original}}{2} \), because \( v_{new} = 2v_{original} \).

The fraction of the hose opening that was blocked can be determined by comparing the new area to the original area. The blocked fraction is given by \( 1 - \frac{A_{new}}{A_{original}} \). Substituting \( A_{new} = \frac{A_{original}}{2} \), the blocked fraction becomes \( 1 - \frac{1}{2} = \frac{1}{2} \).

Conclude that you blocked half of the hose opening to achieve the increased range of the water stream.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Bernoulli's Principle

Bernoulli's Principle states that in a flowing fluid, an increase in the fluid's speed occurs simultaneously with a decrease in pressure or potential energy. This principle helps explain how blocking part of the hose opening increases the velocity of the water exiting the hose, allowing it to reach a greater distance.

Continuity Equation

The Continuity Equation in fluid dynamics asserts that the mass flow rate must remain constant from one cross-section of a pipe to another. This means that if the cross-sectional area of the hose is reduced by blocking it, the speed of the water must increase to maintain the same flow rate, which is crucial for understanding how the water's reach changes.

Projectile Motion

Projectile Motion refers to the motion of an object that is thrown or projected into the air, influenced only by gravity and its initial velocity. In this scenario, the water behaves like a projectile after leaving the hose, and its horizontal distance is affected by the initial speed, which is increased by blocking the hose opening.