Fishy Physics: Calculating Velocity After A Fishy Feast!

Let's dive into a fascinating physics problem involving a hungry fish! This scenario combines concepts of momentum and velocity, giving us a practical example of how these principles work in the real world—or at least, in a hypothetical fish-eat-fish world. We're going to figure out the velocity of a larger fish after it decides to have a smaller fish for lunch. Get ready to explore the depths of momentum conservation!

Understanding the Problem: A Fishy Collision

Picture this: A 3.30 kg fish is cruising along at 4.00 m/s towards the East. Suddenly, it spots a smaller, perhaps less fortunate, 0.530 kg fish swimming South at 4.00 m/s. In a split second, the larger fish makes a meal out of the smaller one. The question we need to answer is: What is the velocity of the larger fish immediately after this underwater feast?

This isn't just about hungry fish; it's about understanding how momentum is conserved in a collision. In this case, the "collision" is the larger fish eating the smaller fish. Since there are no external forces acting on the system (we're ignoring water resistance and other complexities), the total momentum of the system before the event must equal the total momentum after the event. Let's break down the key concepts and then crunch the numbers.

Key Concepts: Momentum and Conservation

• Momentum: Momentum (${p}$) is a measure of an object's mass in motion. It is calculated as the product of an object's mass (${m}$) and its velocity (${v}$): ${p = mv}$. Momentum is a vector quantity, meaning it has both magnitude and direction.
• Conservation of Momentum: The law of conservation of momentum states that the total momentum of an isolated system remains constant if no external forces act on it. In simpler terms, in a closed system, momentum is neither lost nor gained; it is only transferred. This principle is crucial for solving collision problems.

Breaking Down the Problem

1. Initial Momentum of the Larger Fish: The larger fish has a mass of 3.30 kg and is moving East at 4.00 m/s. Therefore, its initial momentum (${p_1}$) can be calculated as: ${p_1 = (3.30 \text{ kg}) \times (4.00 \text{ m/s}) = 13.2 \text{ kg m/s East}}$
2. Initial Momentum of the Smaller Fish: The smaller fish has a mass of 0.530 kg and is moving South at 4.00 m/s. Its initial momentum (${p_2}$) is: ${p_2 = (0.530 \text{ kg}) \times (4.00 \text{ m/s}) = 2.12 \text{ kg m/s South}}$
3. Total Initial Momentum: The total initial momentum of the system is the vector sum of the individual momenta. Since the fishes are moving in perpendicular directions (East and South), we can treat these momenta as components of a right triangle. The total initial momentum (${p_{\text{total, initial}}}$) is the vector sum of ${p_1}$ and ${p_2}$: ${p_{\text{total, initial}} = p_1 + p_2}$
4. Final Momentum of the Combined Fish: After the larger fish eats the smaller fish, they become a single mass. The combined mass (${m_{\text{total}}}$) is: ${m_{\text{total}} = 3.30 \text{ kg} + 0.530 \text{ kg} = 3.83 \text{ kg}}$ Let the final velocity of the combined fish be ${v_{\text{final}}}$ The final momentum (${p_{\text{total, final}}}$) is: ${p_{\text{total, final}} = m_{\text{total}} \times v_{\text{final}}}$
5. Applying Conservation of Momentum: According to the conservation of momentum, the total initial momentum equals the total final momentum: ${p_{\text{total, initial}} = p_{\text{total, final}}}$ Therefore: ${p_1 + p_2 = m_{\text{total}} \times v_{\text{final}}}$

Calculating the Final Velocity

Now that we have all the components, let's calculate the final velocity of the larger fish after its meal. Remember, momentum is a vector quantity, so we'll need to consider both the x (East) and y (South) components.

Breaking Down the Components

1. X-Component (East):
   • Initial momentum in the x-direction: ${p_{1x} = 13.2 \text{ kg m/s}}$
   • Final momentum in the x-direction: ${p_{\text{final, x}} = (3.83 \text{ kg}) \times v_{\text{final, x}}}$
   • Conserving momentum in the x-direction: ${13.2 \text{ kg m/s} = (3.83 \text{ kg}) \times v_{\text{final, x}}}$ ${v_{\text{final, x}} = \frac{13.2 \text{ kg m/s}}{3.83 \text{ kg}} = 3.45 \text{ m/s}}$
2. Y-Component (South):
   • Initial momentum in the y-direction: ${p_{2y} = 2.12 \text{ kg m/s}}$
   • Final momentum in the y-direction: ${p_{\text{final, y}} = (3.83 \text{ kg}) \times v_{\text{final, y}}}$
   • Conserving momentum in the y-direction: ${2.12 \text{ kg m/s} = (3.83 \text{ kg}) \times v_{\text{final, y}}}$ ${v_{\text{final, y}} = \frac{2.12 \text{ kg m/s}}{3.83 \text{ kg}} = 0.553 \text{ m/s}}$

Combining the Components

We now have the x and y components of the final velocity. To find the magnitude and direction of the final velocity, we use the Pythagorean theorem and trigonometry.

1. Magnitude of Final Velocity: ${|v_{\text{final}}| = \sqrt{v_{\text{final, x}}^2 + v_{\text{final, y}}^2}}$ ${|v_{\text{final}}| = \sqrt{(3.45 \text{ m/s})^2 + (0.553 \text{ m/s})^2}}$ ${|v_{\text{final}}| = \sqrt{11.9025 + 0.305809} \text{ m/s}}$ ${|v_{\text{final}}| = \sqrt{12.208309} \text{ m/s}}$ ${|v_{\text{final}}| = 3.49 \text{ m/s}}$
2. Direction of Final Velocity: The direction (${\theta}$) can be found using the arctangent function: ${\theta = \arctan\left(\frac{v_{\text{final, y}}}{v_{\text{final, x}}}\right)}$ ${\theta = \arctan\left(\frac{0.553 \text{ m/s}}{3.45 \text{ m/s}}\right)}$ ${\theta = \arctan(0.1603)}$ ${\theta = 9.11^\circ}$

The Final Answer

After the larger fish consumes the smaller fish, the final velocity of the larger fish is approximately 3.49 m/s at an angle of 9.11° South of East. This means the fish continues to move primarily Eastward, but with a slight southward deflection due to the smaller fish's initial momentum. Isn't physics delicious?

Practical Implications

While this problem might seem purely academic, it illustrates principles applicable in various real-world scenarios. For example, consider collisions in sports (like a tackle in football), the movement of celestial bodies, or even the behavior of particles in a nuclear reaction. Understanding momentum conservation allows engineers and scientists to predict outcomes and design systems more effectively.

Conclusion

So, there you have it! A hungry fish, a quick meal, and the laws of physics combine to give us a fascinating problem to solve. By applying the principle of momentum conservation and breaking down the problem into components, we've determined the final velocity of the larger fish. Remember, physics is all around us, even in the depths of the ocean! Keep exploring, keep questioning, and keep enjoying the wonders of the universe, guys!