Problem 26 - College Physics

Question

To protect their young in the nest, peregrine falcons will fly into birds of prey (such as ravens) at high speed. In one such episode, a \(600 \mathrm{~g}\) falcon flying at \(20.0 \mathrm{~m} / \mathrm{s}\) flew into a \(1.5 \mathrm{~kg}\) raven flying at \(9.0 \mathrm{~m} / \mathrm{s}\). The falcon hit the raven at right angles to its original path and bounced back with a speed of \(5.0 \mathrm{~m} / \mathrm{s} .\) By what angle did the falcon change the raven's direction of motion?

Step-by-Step Solution

Answer

The angle by which the raven's direction changed is approximately 47.7 degrees.

Step 1: Convert Mass Units

First, ensure all mass units are consistent. Convert the falcon's mass from grams to kilograms: \[ 600 \text{ g} = 0.6 \text{ kg} \]

Step 2: Understand the Problem

We have a collision involving two birds: a falcon and a raven. The falcon changes direction after the collision. We need to find the angle at which the raven's direction changes due to the impact.

Step 3: Identify Variables

The falcon's velocity before the collision is \(v_{f1} = 20.0\, \mathrm{m/s}\), and after the collision, it is \(v_{f2} = -5.0\, \mathrm{m/s}\) (changed direction). The raven’s initial velocity is \(v_{r1} = 9.0\, \mathrm{m/s}\). The masses are \( m_f = 0.6\, \mathrm{kg} \) and \( m_r = 1.5\, \mathrm{kg} \). The raven's speed after the collision is unknown, \(v_{r2}\), and we want to find the angle change.

Step 4: Calculate Falcon's Momentum Change

Calculate the change in momentum of the falcon in the x-direction (assuming the original path is along the x-axis):\[ \Delta p_f = m_f \times (v_{f2} - v_{f1}) = 0.6 \times (-5.0 - 20.0) = -15 \text{ kg m/s} \]

Step 5: Apply Conservation of Momentum

Since momentum is conserved and directions are perpendicular, the raven's change in momentum is equal in magnitude and opposite in direction to the falcon's change in momentum.Let \(p_{r2x}\) and \(p_{r2y}\) be the x and y components of the raven's final momentum. The conservation equations are:\[ m_r \times v_{r1} = p_{r2x} \]\[ 0 = p_{r2y} - 15 \]

Step 6: Solve Momentum Equations

Calculate \(p_{r2x}\):\[ 1.5 \times 9.0 = p_{r2x} \Rightarrow p_{r2x} = 13.5 \text{ kg m/s} \]Calculate \(p_{r2y}\):\[ p_{r2y} = 15 \text{ kg m/s} \]

Step 7: Calculate the Raven's Speed and Change in Angle

Use the Pythagorean theorem to find the raven's final speed, \(v_{r2}\):\[ p_{r2} = \sqrt{p_{r2x}^2 + p_{r2y}^2} = \sqrt{13.5^2 + 15^2} \]Find the tangent of the angle, \(\theta\), using the components:\[ \tan \theta = \frac{p_{r2y}}{p_{r2x}} = \frac{15}{13.5} \]Calculate \(\theta\) using the inverse tangent function.

Step 8: Calculate Angle Change

Compute \(\theta\):\[ \theta = \tan^{-1} \left( \frac{15}{13.5} \right) \]Calculate the angle in degrees to find the angle by which the raven's direction of motion changed.

Key Concepts

Collision Problems

Collision problems are a classic example of how physics applies to real-world interactions. When two objects collide, they exchange energy and momentum, resulting in changes to their velocities and directions. In our specific scenario, a peregrine falcon collides with a raven, significantly impacting both of their flight paths.

To solve collision problems, it is essential to understand the principle of momentum conservation. In the absence of external forces, the total momentum before the collision equals the total momentum after the collision. This means that the changes in momentum for the colliding objects must balance each other out. In our problem, we look at both falcon and raven's momentum, observing how the falcon's impact re-routes the raven's flight trajectory.

Paying attention to the direction and speed before and after the collision helps in calculating forces and eventual paths. Consider all variables, like before and after velocities of both objects, to map out the post-collision behavior and determine angles.

Physics Problem Solving

Physics problem solving requires a clear methodology and understanding of the fundamental concepts applied. For exercises dealing with collisions, it's often necessary to work through the problem systematically.

Start by identifying and converting any units as necessary to ensure all measurements are in compatible forms. Before diving into equations, clearly define your known variables: initial velocities, masses, and the post-collision behaviors of the objects in question.

Utilize relevant physics laws like the law of conservation of momentum. For this problem, we needed to equate the total momentum before and after the collision in each coordinate direction. Make sure to break complex motions into components, which often means separating movements into perpendicular directions, like x and y-axis. By managing these smaller, simpler sub-problems independently and then combining their results, we can solve most physics puzzles with confidence.

Angle of Deflection

Understanding how and why angles change in a collision can provide deep insights into motions. In a scenario where two objects collide, the angle of deflection measures the degree to which one object changes direction due to impact. For the raven impacted by the falcon, this deflection angle indicates how drastically the raven's path is altered.

Calculation involves decomposing the resulting motion into x and y components and using trigonometric relationships. Once we have the momentum components, apply the tangent function: the tangent of the angle between the post-collision path and the original path is the ratio of the y-component to the x-component of momentum.

Finally, perform the inverse tangent function to find the deflection angle. The calculated angle will be the degree measure of how far the raven's path has deviated due to the collision. This explanation of vector components into angles demonstrates how physics can quantify what at first seems qualitative.

Peregrine Falcon Behavior

Peregrine falcons are remarkable birds of prey, renowned for their extraordinary speeds and precision in hunting. One notable behavior involves protecting their nests aggressively, often by colliding with larger birds that pose threats to their offspring.

Their ability to reach high speeds, sometimes over 300 km/h during a dive, allows falcons to surprise and redirect intruders effectively. This agility and strength make them unique among aerial predators. In our exercise, this capability is mathematically depicted as the falcon alters the course of a much larger raven through a collision.

Understanding the behavior of peregrine falcons in scientific contexts, such as physics problems, showcases their physical adaptations and how nature can be represented and understood through equations and momentum principles.