Problem 27 - College Physics

Question

\(\bullet\) A hungry 11.5 \(\mathrm{kg}\) predator fish is coasting from west to east to east at 75.0 \(\mathrm{cm} / \mathrm{s}\) when it suddenly swallows a 1.25 \(\mathrm{kg}\) fish swim- ming from north to south at 3.60 \(\mathrm{m} / \mathrm{s} .\) Find the magnitude and direction of the velocity of the large fish just after it snapped up this meal. Neglect any effects due to the drag of the water.

Step-by-Step Solution

Answer

The final velocity is 0.764 m/s at 27.9° south of east.

Step 1: Understand Conservation of Momentum

In this problem, we need to apply the principle of conservation of momentum because the predator fish and its prey can be treated as an isolated system (negligible effects of water drag). The total momentum before the predator swallows its prey is equal to the total momentum just after.

Step 2: Calculate Initial Momentum Components

Calculate the momentum of each fish in both the east and south directions. The predator fish is moving east with a velocity of 75.0 cm/s (or 0.75 m/s), so its momentum in the x-direction is:\[ p_{predator,x} = m_{predator} \times v_{predator,x} = 11.5 \, \text{kg} \times 0.75 \, \text{m/s} = 8.625 \, \text{kg m/s} \]The smaller fish is swimming south, so its momentum in the y-direction is:\[ p_{prey,y} = m_{prey} \times v_{prey,y} = 1.25 \, \text{kg} \times 3.60 \, \text{m/s} = 4.5 \, \text{kg m/s} \]

Step 3: Apply Conservation of Momentum to Find Final Velocity Components

After the predator swallows the prey, both will move together. Thus, the final momentum components in the x and y directions are:\[ p_{final,x} = p_{predator,x} = 8.625 \, \text{kg m/s} \]\[ p_{final,y} = p_{prey,y} = 4.5 \, \text{kg m/s} \]The total mass of the combined system is:\[ m_{total} = m_{predator} + m_{prey} = 11.5 \, \text{kg} + 1.25 \, \text{kg} = 12.75 \, \text{kg} \]Thus, the velocity components after the event are given by:\[ v_{final,x} = \frac{p_{final,x}}{m_{total}} = \frac{8.625}{12.75} = 0.6765 \, \text{m/s} \]\[ v_{final,y} = \frac{p_{final,y}}{m_{total}} = \frac{4.5}{12.75} = 0.3529 \, \text{m/s} \]

Step 4: Calculate Magnitude and Direction of Final Velocity

The magnitude of the final velocity can be calculated using Pythagorean theorem:\[ v_{final} = \sqrt{v_{final,x}^2 + v_{final,y}^2} = \sqrt{(0.6765)^2 + (0.3529)^2} = 0.764 \, \text{m/s} \]The direction (angle \( \theta \) from the east towards south) can be calculated by:\[ \theta = \arctan{\left(\frac{v_{final,y}}{v_{final,x}}\right)} = \arctan{\left(\frac{0.3529}{0.6765}\right)} \approx 27.9^\circ \]

Key Concepts

Velocity Calculation

In physics, calculating velocity is essential for understanding motion. Velocity is a vector quantity, which means it has both magnitude and direction.
For scenarios involving multiple objects, like in our fish example, we find individual velocity components first. Velocity components are typically separated into perpendicular directions, such as east-west for the x-axis and north-south for the y-axis.
By computing these components separately, we can easily understand how each object contributes to the overall motion. After finding the components, we use them to determine the resultant velocity using the Pythagorean theorem. This theorem applies because the velocity components form a right-angled triangle.
Thus, the magnitude of velocity is:

• \( v=\sqrt{v_{x}^2 + v_{y}^2} \)

Momentum Components

Momentum is another vector quantity, similar to velocity. It is derived from the product of an object’s mass and its velocity. In the case of our two fish, each has momentum in their respective moving directions.
Breaking momentum into components makes it easier to handle complex interactions, especially when applied on different paths. For example:

• Eastward or x-direction: predator's momentum
• Southward or y-direction: prey's momentum

In the x-direction, the conservation of momentum allows us to retain the total eastward momentum before and after the event. Similarly, in the y-direction, the southward momentum remains constant in an isolated system.
These components are crucial for predicting the system's behavior when combined as a single entity. Calculating these accurately can give insights into resultant effects and help in precise determination of motion.

Isolated System

An isolated system is one where no external forces affect the internal components. In physics, assuming a system is isolated simplifies many calculations because we can ignore factors like friction, drag, or external interventions.
In our fish scenario, since water drag is neglected, the predator and prey fish form an isolated system for the conservation of momentum. This condition holds true because only internal forces (collision and swallowing) are considered, and they cannot change the system’s total momentum independently.
The concept of an isolated system ensures that the momentum observed before an interaction stays constant afterward, aiding in the calculation of final states like velocity and direction.

Momentum Conservation Principle

The principle of momentum conservation is a fundamental law of physics stating that if no external forces act on a system, the system's total momentum remains constant. This principle applies to our fish example because the effect of water resistance is negligible, allowing us to treat the system as closed.
In practice, how does this apply to our fish? Before the hungry predator captures its prey, the total momentum is the sum of their individual momentums in their respective directions. After swallowing, the momentum of the combined system remains unchanged.
Thus, using the equation:

• Pre-capture momentum = Post-capture momentum

This shows that whatever happens inside this system, the total momentum stays the same. It is a key tool used in predicting outcomes after collisions or mergers, like in our fish example.
Understanding this concept is critical to analyze and predict interactions in similar physical scenarios effectively.