Problem 72 - College Physics

Question

A \(15.0 \mathrm{~g}\) acorn falls from rest from the top of a \(35.0-\mathrm{m}\) -high oak tree. When it is halfway to the ground, a \(135 \mathrm{~g}\) bird gliding horizontally at \(75.0 \mathrm{~cm} / \mathrm{s}\) scoops it up with its beak. Find (a) the horizontal and vertical components of the bird's velocity, and (b) the speed of the bird and the angle its velocity makes with the vertical, just after the bird scoops up the acorn.

Step-by-Step Solution

Answer

Horizontal velocity after event: 0.693 m/s, Vertical velocity: 1.928 m/s, Speed: 2.052 m/s, Angle: 19.44° from vertical.

Step 1: Understand the Problem

We are given an acorn falling from a tree and a bird that catches it while flying horizontally. We need to find the resulting velocity of the bird and the acorn after this event.

Step 2: Calculate Acorn's Velocity Before Collision

The acorn falls from rest, so we use the equation for the velocity of a freely falling object: \[ v = \sqrt{2gh} \]where \( g = 9.81 \, m/s^2 \) and \( h = 17.5 \, m \) (since the acorn has fallen halfway). Plug in the values: \[ v = \sqrt{2 \times 9.81 \, m/s^2 \times 17.5 \, m} \approx 18.51 \, m/s \].

Step 3: Bird's Initial Horizontal Velocity

The bird is gliding horizontally, so its initial horizontal velocity is given as \( 75.0 \, cm/s \), which is \( 0.75 \, m/s \).

Step 4: Use Conservation of Momentum

The system to consider is the bird plus acorn as they collide and move together. Apply the conservation of momentum to both components:- Horizontal: \( m_{b}v_{b,x} + m_{a}v_{a,x} = (m_{b} + m_{a})v_{fx} \)- Vertical: \( m_{b}v_{b,y} + m_{a}v_{a,y} = (m_{b} + m_{a})v_{fy} \)Where:- \( m_{b} = 0.135 \, kg \) (mass of the bird)- \( v_{b,x} = 0.75 \, m/s \) (initial horizontal velocity of the bird)- \( v_{b,y} = 0 \, m/s \) (initial vertical velocity of the bird)- \( m_{a} = 0.015 \, kg \) (mass of the acorn)- \( v_{a,x} = 0 \, m/s \) (initial horizontal velocity of the acorn)- \( v_{a,y} = 18.51 \, m/s \) (downward velocity of the acorn)Solving for \( v_{fx} \) and \( v_{fy} \):\( v_{fx} = \frac{0.135 \, kg \times 0.75 \, m/s + 0.015 \, kg \times 0 \, m/s}{0.135 \, kg + 0.015 \, kg} \approx 0.693 \, m/s \)\( v_{fy} = \frac{0.135 \, kg \times 0 \, m/s + 0.015 \, kg \times 18.51 \, m/s}{0.135 \, kg + 0.015 \, kg} \approx 1.928 \, m/s \)

Step 5: Calculate the Bird's Speed and Angle

The resultant speed is calculated using the Pythagorean theorem: \[ v_f = \sqrt{v_{fx}^2 + v_{fy}^2} \approx \sqrt{0.693^2 + 1.928^2} = 2.052 \, m/s \]The angle with the vertical is given by: \[ \theta = \tan^{-1}\left(\frac{v_{fx}}{v_{fy}}\right) \approx \tan^{-1}\left(\frac{0.693}{1.928}\right) \approx 19.44^\circ \]

Key Concepts

Velocity Calculation

In physics, calculating velocity is crucial for understanding how fast an object is moving and in which direction. Velocity is a vector quantity, meaning it has both magnitude (speed) and direction. To compute velocity in scenarios involving both horizontal and vertical motion, you often need to resolve the movement into components.

For the acorn problem, the first step in calculating its velocity involved using a formula from the Physics of free fall: \[ v = \sqrt{2gh} \]Here, \(g\) is the acceleration due to gravity, and \(h\) is the height fallen. This formula helps find the vertical component of velocity, crucial when the acorn is influenced purely by gravity.

For the bird, which is initially gliding horizontally, calculating its velocity involves a much simpler process since it's already given its horizontal speed: 0.75 m/s.

Free Fall Motion

Free fall is a specific type of motion where gravity is the only force acting upon an object. This results in the acceleration toward the earth being constant, at approximately \(9.81 \, m/s^2\). In the acorn exercise, free fall principles help determine the speed and impact of the acorn as it falls halfway to the ground from the initial 35 meters.

Since the acorn starts from rest, its initial velocity is zero. To find its speed at halfway (17.5 meters), we used the free fall formula \( v = \sqrt{2gh} \).It calculated roughly to 18.51 m/s. This is solely vertical since free fall doesn't involve horizontal motion unless external forces act on it.
Observing this helps highlight how objects behave under gravity and aids in predicting trajectories and impact in various situations.

Horizontal and Vertical Components

Understanding the horizontal and vertical components of motion is essential for analyzing complex movements. When an object moves, its velocity can often be split into these components, simplifying the calculations and understanding of its trajectory.

For the bird-acorn problem, after the encounter, the separate motions combine:

• Horizontal component: The bird's initial horizontal velocity is presented directly as 0.75 m/s, since it glides horizontally.
• Vertical component: The acorn's vertical speed due to free fall is 18.51 m/s. The bird's vertical speed initially is zero, as it moves horizontally.

By employing the conservation of momentum separately for each component, you determine the resultant combined horizontal and vertical velocities: \[ v_{fx} = 0.693 \, m/s, \quad v_{fy} = 1.928 \, m/s\]Knowing these allows for further calculations relating to the overall speed and angle of motion.

Angle of Trajectory

The angle of trajectory details the path inclination related to the vertical or horizontal axis. Understanding this angle is key for predicting where and how an object will move after an event or collision.

In this scenario, after the bird captures the acorn, the combined entity moves at an angle. Using:\[\theta = \tan^{-1}\left(\frac{v_{fx}}{v_{fy}}\right)\]you find the angle of motion post-collision. Here, it computes to approximately 19.44°, with respect to the vertical.
This provides an insight into the resulting path of the bird-acorn system, indicating a slight deviation from the primarily vertical path due to the preserved horizontal momentum of the bird.