Question

A 15.0-kg fish swimming at 1.10 m/s suddenly gobbles up a 4.50-kg fish that is initially stationary. Ignore any drag effects of the water. (a) Find the speed of the large fish just after it eats the small one. (b) How much mechanical energy was dissipated during this meal?

Short answer

(a) 0.846 m/s; (b) 2.111 J of energy dissipated.

Step-by-step solution

Understand the Problem

We have a two-fish system where a larger fish of mass 15.0 kg and velocity 1.10 m/s swallows a smaller, stationary fish of mass 4.50 kg. We need to find the speed of the large fish after the event and the energy dissipated in the process.

Apply Conservation of Momentum

Using the law of conservation of momentum, which states that the total momentum before an event is equal to the total momentum afterward, we set up the equation. The initial momentum of the system is the product of the mass and velocity of the large fish, since the small one is stationary: \( p_i = m_1 v_1 + m_2 v_2 = 15.0 \text{ kg} \times 1.10 \text{ m/s} + 4.50 \text{ kg} \times 0 \text{ m/s} \). After gobbling, the system becomes a single mass of two fishes together moving at a velocity \( v_f \): \( p_f = (m_1 + m_2) \times v_f \). Setting these equal \( m_1 v_1 = (m_1 + m_2) v_f \).

Calculate Final Velocity

Let's solve for \( v_f \): \( 15.0 \text{ kg} \times 1.10 \text{ m/s} = (15.0 \text{ kg} + 4.50 \text{ kg}) \times v_f \). The final equation is \( 16.5 \text{ kg} \cdot \text{m/s} = 19.5 \text{ kg} \times v_f \). Solving for \( v_f \), we get \( v_f = \frac{16.5}{19.5} \text{ m/s} = 0.846 \text{ m/s} \).

Calculate Initial and Final Kinetic Energy

Initial kinetic energy \( KE_i \) is calculated using \( KE = 0.5 \times m \times v^2 \). For the large fish, \( KE_i = 0.5 \times 15.0 \text{ kg} \times (1.10 \text{ m/s})^2 = 9.075 \text{ J} \). The small fish's initial kinetic energy is zero since it is stationary. Total initial kinetic energy = 9.075 J.

Calculate Final Kinetic Energy of the Combined Fish

After swallowing the small fish, the mass of the combined fish is 19.5 kg, moving at 0.846 m/s. Calculate the final kinetic energy \( KE_f = 0.5 \times 19.5 \text{ kg} \times (0.846 \text{ m/s})^2 = 6.964 \text{ J} \).

Calculate Mechanical Energy Dissipated

Energy dissipated is the difference between the initial and final kinetic energies: \( \text{Energy Gained} = \text{Initial KE} - \text{Final KE} = 9.075 \text{ J} - 6.964 \text{ J} = 2.111 \text{ J} \). Thus, 2.111 J of energy was dissipated as mechanical energy.

Key concepts

Mechanical Energy Dissipation

When we talk about mechanical energy dissipation, we're referring to the energy that "disappears" or is transformed into other forms like heat or sound during a physical process. In the context of our fish problem, when the larger fish swallows the smaller one, some of the system's mechanical energy is lost. This energy doesn't just vanish; it generally converts into forms that we don't typically measure in the system's kinetic or potential energy.

In our scenario, the initial kinetic energy of the system is higher than the final kinetic energy. The difference here, found to be 2.111 Joules, is the energy that has dissipated. This dissipation is common in real-world processes due to friction, deformation, or, as in this case, absorption during the event of collision or consumption. Such calculations are essential in physics to understand energy efficiency in systems.

• Energy dissipation shows the inefficiency of energy transfer.
• Recognizing dissipated energy helps in designing systems with less energy loss.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion. It's an essential concept in physics and is calculated using the formula: \[ KE = \frac{1}{2}mv^2 \]where \(m\) is mass and \(v\) is velocity. In our exercise, before the large fish eats the smaller stationary fish, only the larger fish has kinetic energy.

Initially, the large fish's kinetic energy was 9.075 Joules. However, once it swallows the small fish, the entire system slows down slightly, resulting in a new lower kinetic energy of 6.964 Joules. This reduction reflects that some kinetic energy got transformed into other forms during the interaction.

• Kinetic energy depends significantly on velocity.
• Even small changes in speed can lead to significant changes in kinetic energy due to the square dependence.
• Understanding kinetic energy provides insight into how and why objects move or stop.

Physics Problem Solving

Effective problem solving in physics requires a clear understanding and application of fundamental principles. Here, we utilized the Law of Conservation of Momentum. This principle tells us that in a closed system without external forces, the total momentum remains constant.

To solve the fish problem, understanding momentum allowed us to find the new speed of the system after the event. We set the product of mass and velocity before the event equal to the product after, because momentum should stay the same if isolated.

• Identifying known values: masses and initial velocities.
• Applying relevant concepts: momentum conservation for velocity and kinetic energy calculations.
• Consistent unit usage ensures accurate calculations.
• Checking units provides insight and can catch errors early in problem-solving.