Problem 75 - College Physics

Question

Archerfish are tropical fish that hunt by shooting drops of water from their mouths at insects above the water's surface to knock them into the water, where the fish can eat them. \(\mathrm{A} 65 \mathrm{~g}\) fish at rest just on the water's surface can expel a \(0.30 \mathrm{~g}\) drop of water in a short burst of \(5.0 \mathrm{~ms}\). High-speed measurements show that the water has a speed of \(2.5 \mathrm{~m} / \mathrm{s}\) just after the archerfish expels it. What is the momentum of one drop of water immediately after it leaves the fish's mouth? A. \(7.5 \times 10^{-4} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\) B. \(1.5 \times 10^{-4} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\) C. \(7.5 \times 10^{-3} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\) D. \(1.5 \times 10^{-3} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\)

Step-by-Step Solution

Answer

The correct answer is option A: \(7.5 \times 10^{-4} \text{ kg} \cdot \text{m/s}\).

Step 1: Understand the Problem

We need to find the momentum of the expelled water drop immediately after it leaves the fish's mouth. The momentum can be calculated using the formula \( p = m imes v \), where \( p \) is the momentum, \( m \) is the mass, and \( v \) is the velocity of the object.

Step 2: Identify the Variables

Here, the mass \( m \) of the water drop is \(0.30\text{ g} = 0.30/1000\text{ kg} = 0.0003\text{ kg}\) and its velocity \( v \) is \(2.5\text{ m/s}\).

Step 3: Use the Momentum Formula

Using the momentum formula \( p = m imes v \): \[p = 0.0003\, \text{kg} \times 2.5\, \text{m/s}\]

Step 4: Perform the Calculation

Calculate the product to find the momentum: \[p = 0.0003 \times 2.5 = 0.00075\, \text{kg} \cdot \text{m/s}\] So, the momentum of the drop is \(0.00075\, \text{kg} \cdot \text{m/s}\).

Step 5: Choose the Correct Answer

The calculated momentum \(0.00075\, \text{kg} \cdot \text{m/s}\) matches option A: \(7.5 \times 10^{-4} \text{ kg} \cdot \text{m/s}\). Thus, option A is the correct choice.

Key Concepts

Archerfish

Archerfish are fascinating creatures that live in tropical waters. They have a unique hunting strategy that sets them apart from many other fish.
These fish have the remarkable ability to shoot drops of water from their mouths with precision to knock insects into the water.
This unique hunting method is both incredible and effective, allowing archerfish to feed on insects that would otherwise be out of their reach.

• Archerfish use their sharp eyesight to spot prey above the water.
• They calculate the right angle and amount of force needed to hit the target.
• The fish need to account for light refraction, as water can bend light, distorting the position of the prey.

This skill involves complex physics and gives the fish a distinct advantage in their environment.

Conservation of Momentum

The principle of conservation of momentum is central to understanding what happens when the archerfish shoots a droplet of water.
In physics, momentum is defined as the product of an object's mass and its velocity.
Conservation of momentum tells us that in a closed system, the total momentum remains constant unless acted on by an external force. This is particularly useful for solving problems involving interactions between objects, like when the archerfish spits water.

• The fish and the water droplet form a system where momentum is exchanged, but the total momentum remains the same.
• When the fish expels water, it experiences a backward motion, an example demonstrating momentum conservation.
• Understanding this concept helps predict the direction and speed of objects post-interaction.

This fundamental principle helps us not only in simple physics problems but also in more complex situations like rocket propulsion.

Physics Problems

Solving physics problems, like the one involving the archerfish, involves a logical, step-by-step approach. Understanding how to break down complex scenarios into manageable parts is crucial.
To tackle such problems effectively, one must first identify known quantities like mass and velocity before applying relevant physics principles.
Here's how you can approach physics problems:

• Read the problem carefully to understand what's being asked.
• Identify the key variables and write them clearly.
• Choose the appropriate physical principles or formulas that apply.
• Perform the necessary calculations accurately.
• Verify the answer against multiple choice or expected results.

Practicing these steps enhances problem-solving skills and builds a strong foundation in physics.

Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the forces causing the motion.
In the context of the archerfish, kinematics comes into play when we calculate the velocity of the water drop as it leaves the fish's mouth.
Kinematics involves several core concepts:

• Displacement: The change in position of an object.
• Velocity: The speed of an object in a given direction. The velocity of the water droplet in our problem is a key variable.
• Acceleration: The change in velocity over time (though not a focus in this particular problem).

These concepts are the foundation for describing how objects move and predicting future motion. Understanding kinematics allows us to quantify and predict the behavior of moving bodies, converting complex scenarios into understandable calculations.