Problem 8 - Applied Physics

Question

A garden hose is used to fill a bucket in \(30.0 \mathrm{~s}\). If you cover part of the hose nozzle so the speed of the water leaving the hose doubles, how long does it take to fill the bucket?

Step-by-Step Solution

Answer

The bucket will fill in 15 seconds when the speed of water doubles.

Step 1: Understanding the Problem

We need to determine how long it will take to fill a bucket when the speed of water from the hose doubles. We know that initially, the bucket takes 30 seconds to fill with a certain speed.

Step 2: Initial Time to Fill the Bucket

The initial time given to fill the bucket is 30 seconds at speed \(v\).

Step 3: Effect of Doubling the Speed

If the speed of water doubles, the flow rate (volume/time) also doubles because the flow rate is directly proportional to speed.

Step 4: Calculating the New Time

Since the flow rate is doubled, the time to fill the bucket reduces by half. Therefore, divide the initial time by 2 to get the new time.

Step 5: Final Calculation

The new time taken to fill the bucket is \(\frac{30}{2} = 15\) seconds.

Key Concepts

Flow Rate

Flow rate is a key concept in fluid dynamics that helps us understand how quickly a fluid is moving. It is often expressed as the volume of fluid that passes a particular point in a given amount of time. In the example of filling a bucket with a garden hose, the flow rate would be the amount of water exiting the hose per second.

The formula for flow rate is given by the equation: \[Q = A \times v\] where \(Q\) is the flow rate, \(A\) is the cross-sectional area of the hose opening, and \(v\) is the speed of the water. By understanding this relationship, we can determine how changes in speed, such as doubling the speed, would impact the flow rate.

Speed and Time Relationship

The relationship between speed and time is crucial in determining how quickly an event occurs. This is particularly true in scenarios involving fluid flow, like with our garden hose example.

In basic terms, if the speed of a fluid increases, the time taken for a given volume of that fluid to reach a destination decreases. This shift can be visualized by considering that a bucket filled by a hose pressurized to deliver water at high speed would fill faster than if the water trickled slowly. The relationship is inversely proportional, meaning as one increases, the other decreases if the volume remains constant.

Doubling Speed Effect

When you double the speed of the flow, an interesting set of changes occurs. By doubling the flow speed, you’re effectively doubling the flow rate, assuming the cross-sectional area of the flow remains unchanged.

For the garden hose, doubling the speed implies that water exits the nozzle twice as fast, leading to a bucket filling in half the time it would initially. Hence, if the original time to fill was 30 seconds, with doubled speed, it reduces to 15 seconds. This is because the flow rate—volume per time unit—has increased, thus reducing the time required to fill the bucket with the same volume of water.

Proportional Reasoning

Proportional reasoning is a mathematical tool that helps us understand relationships between variables like speed, time, and flow rate.

In this particular problem, the understanding that doubling the speed also doubles the flow rate is rooted in proportional reasoning. By realizing that speed and flow rate are directly proportional, we can deduce the effect on time, which is inversely proportional to flow rate. This principle not only helps in this problem but is widely applicable in many real-world situations where understanding the relationship between changing quantities is essential. By practicing proportional reasoning, students can enhance their problem-solving skills, making complex problems simpler.