Problem 76 - 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 speed of the archerfish immediately after it expels the drop of water? A. \(0.0025 \mathrm{~m} / \mathrm{s}\) B. \(0.012 \mathrm{~m} / \mathrm{s}\) C. \(0.75 \mathrm{~m} / \mathrm{s}\) D. \(2.5 \mathrm{~m} / \mathrm{s}\)

Step-by-Step Solution

Answer

The speed of the archerfish immediately after expelling the drop of water is approximately \(0.012\, \text{m/s}\), which corresponds to option B.

Step 1: Identify Initial Conditions

The fish is initially at rest, meaning its initial velocity is zero. The mass of the fish is given as \(65\, \text{g}\) (or \(0.065\, \text{kg}\)) and the mass of the water droplet is \(0.30\, \text{g}\) (or \(0.0003\, \text{kg}\)).

Step 2: Apply Conservation of Momentum

Since the fish initially is at rest and expels the drop of water, the law of conservation of momentum applies. The initial momentum of the system (fish + water) is zero because both the fish and water are at rest before the expulsion. Therefore, the final momentum of the fish and water droplet must also be zero, expressed by:\[m_{fish} \cdot v_{fish} + m_{water} \cdot v_{water} = 0\,\]where \(v_{water} = 2.5\, \text{m/s}\) and \(v_{fish}\) is what we want to find.

Step 3: Solve for Fish's Velocity

Rearrange the momentum formula to solve for the velocity of the fish, \(v_{fish}\):\[v_{fish} = -\frac{m_{water} \cdot v_{water}}{m_{fish}}\]Plug in the values:\[v_{fish}= -\frac{0.0003\, \text{kg} \times 2.5\, \text{m/s}}{0.065\, \text{kg}} = -0.01154\, \text{m/s}\]The negative sign indicates the direction is opposite to that of the water, but the speed is \(0.01154\, \text{m/s}\).

Step 4: Match the Calculated Speed to Given Options

Our calculated speed of the fish is approximately \(0.01154\, \text{m/s}\). Comparing this to the provided options, the closest match is option B: \(0.012\, \text{m/s}\).

Key Concepts

Physics Problem Solving

Physics problem solving involves using logical steps to analyze and solve problems by applying fundamental principles of physics. In solving the archerfish problem, we begin by identifying key elements such as initial conditions and forces involved. The fish and water droplet are initially at rest, allowing us to apply the conservation of momentum. Always start by writing down given data and conditions:
\(\cdot\) Fish mass: \(65\, \text{g} = 0.065\, \text{kg}\)
\(\cdot\) Water drop mass: \(0.30\, \text{g} = 0.0003\, \text{kg}\)
\(\cdot\) Water speed after expulsion: \(2.5\, \text{m/s}\)

Establish what needs to be found, like the speed of the fish after the water is expelled. Then, apply a suitable physics concept or law, such as conservation of momentum, to set up the equation. This systematic approach ensures that you cover each aspect of the problem and helps avoid overlooking crucial details.

Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects, without considering the forces that cause this motion. While solving problems like the archerfish scenario, understanding how movement occurs is essential. Here, we primarily consider:

• Initial velocities: The fish is initially at rest (zero velocity).
• Final velocities: After expelling the water, both the fish and the water have velocities pointing in opposite directions.
• Time of movement: The short burst duration, \(5.0\, \text{ms}\), tells us the action occurs quickly.

In this example, although the forces exerted by the fish are not directly considered, understanding motion concepts helps decipher how the velocities will change. Specifically, the outcome's dependency on mass and velocity aids in predicting the subsequent speed of the fish.

Momentum Calculations

Momentum calculations involve using the principle of momentum, defined as the product of mass and velocity. In physics, momentum is crucial for problems involving moving objects because it helps describe the motion quantitatively. For the archerfish, the formula for conservation of momentum is \[m_{fish} \cdot v_{fish} + m_{water} \cdot v_{water} = 0\]Starting with the known values from the problem and solving for the unknown, \(v_{fish}\):

• Expression rearranged to find fish velocity:
  \[v_{fish} = -\frac{m_{water} \cdot v_{water}}{m_{fish}}\]
• Substitute numbers to get:
  \[v_{fish} = -\frac{0.0003\, \text{kg} \times 2.5\, \text{m/s}}{0.065\, \text{kg}} = -0.01154\, \text{m/s}\]

The result shows that \(v_{fish}\) has a negative sign, which indicates the direction, but the speed itself is \(0.01154\, \text{m/s}\). By comparing this value to given choices in the problem, it matches option B: \(0.012\, \text{m/s}\). This process highlights how momentum calculations are a powerful tool in physics problem-solving contexts.