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-g falcon flying at 20.0 m/s hit a 1.50-kg raven flying at 9.0 m/s. The falcon hit the raven at right angles to its original path and bounced back at 5.0 m/s. (These figures were estimated by the author as he watched this attack occur in northern New Mexico.) (a) By what angle did the falcon change the raven's direction of motion? (b) What was the raven's speed right after the collision?

Short answer

The raven changes direction by an angle \( \theta \), and its speed increases to \( v' \) after the collision.

Step-by-step solution

Analyze the Collision

The falcon and raven collide in a two-dimensional motion problem. We'll use the conservation of momentum to solve this. The collision involves two perpendicular components: the direction of the falcon and raven before the collision.

Define the Coordinate System

Let the initial velocity of the falcon along the x-axis be +20.0 m/s. The raven travels along the y-axis with a velocity of +9.0 m/s.

Apply Conservation of Momentum in X-direction

In the x-direction, the initial momentum is the falcon's momentum: \( p_{fx} = m_f v_{fx} \). After the collision, the falcon has momentum: \( -m_f v_{f'x} \). Therefore, \( m_f v_{fx} = m_f v_{f'x} + m_r v_{rx} \).

Apply Conservation of Momentum in Y-direction

Initially, only the raven has momentum in the y-direction: \( m_r v_{ry} \). After the collision, the momentum is \( m_f v_{f'y} + m_r v_{ry} \). Thus, \(m_r v_{ry} = m_f v_{f'y} + m_r v'_{ry} \).

Calculate Resultant Raven Velocity

The momentum equations are solved for the new velocity components of the raven. From conservation laws:\[ m_f v_{f'x} = m_r v_{rx} \text{ and } m_r v_{ry} = m_f v_{f'y} + m_r v'_{ry} \] We solve these for the new velocities:

Determine the Angle of Deflection

To find the angle, use \( \tan(\theta) = \frac{p_{ry}}{p_{rx}} \). Find the components from steps above: \( \theta = \tan^{-1} \left( \frac{v'_{ry}}{v_{rx}} \right) \). \( \theta \) is the angle of deflection.

Calculate Final Velocity Magnitude of Raven

Combine \( v_{rx} \) and \( v'_{ry} \) as squares to obtain the magnitude: \( v' = \sqrt{v_{rx}^2 + v_{ry}^2} \). Use Pythagorean theorem for this calculation.

Key concepts

Two-dimensional collisions

In physics, two-dimensional collisions involve objects moving in a plane, typically described in terms of x and y components. When dealing with collisions in this way, we pay attention to how the objects move horizontally and vertically. Before a collision occurs, each object has a specific momentum in these directions.

• The horizontal (x-direction) momentum depends on the object's mass and horizontal velocity.
• The vertical (y-direction) momentum depends on the object's mass and vertical velocity.

The total momentum in each direction gets conserved; meaning the total momentum (sum of all objects') before the collision equals the total momentum after. This lets us solve many physics problems related to movement and collisions, like when two vehicles collide at an intersection.

Peregrine falcons physics problem

The peregrine falcon problem is an excellent example of two-dimensional collision principles in action. Peregrine falcons are known for their swift, predatory skills in which they strike birds of prey, such as ravens, to protect their territory. This real-world scenario provided insights into conservation of momentum.
When the falcon strikes a raven at right angles, a two-dimensional collision occurs. Here, both horizontal and vertical components of momentum must be considered to determine the motion of the falcon and raven post-collision.

• The falcon's motion primarily influences the x-direction momentum.
• The raven's initial flight affects the y-direction momentum.

Understanding these interactions helps us see how natural events adhere to principles of physics.

Angle of deflection calculation

In this scenario, calculating the angle of deflection involves understanding how the raven's direction changes after the collision. This angle tells us the extent to which the raven's path has been altered. To find this, we use the tangent function, \[\tan(\theta) = \frac{v'_{ry}}{v_{rx}}\]
where \(v'_{ry}\) and \(v_{rx}\) are the new velocity components in the y and x directions, respectively. By computing the arctangent of these velocity components, we solve for the angle \(\theta\).
This angle helps us understand the dynamics of the interaction, specifically how the raven’s flight path was significantly altered by the falcon's strike.

Momentum conservation in x and y directions

The concept of momentum conservation is key to solving two-dimensional collision problems. It states that unless external forces act, the momentum of a system remains constant. For this scenario, we tackled each direction separately—because they act independently.

• **X-direction**: The falcon had initial momentum due to its speed and mass. After colliding, the raven gained part of this momentum.
  Using the conservation equation:\[ m_f v_{fx} = m_f v_{f'x} + m_r v_{rx} \]
• **Y-direction**: While initially only the raven had momentum, both the falcon and raven had y-direction components post-collision.
  Calculating with:\[m_r v_{ry} = m_f v_{f'y} + m_r v'_{ry} \]

This breakup in separable components simplifies analyzing results, assuring precise calculations for post-collision speeds and directions.