Question

Three birds are flying in a compact formation. The first bird, with a mass of \(100 . g\), is flying \(35.0^{\circ}\) east of north at a speed of \(8.00 \mathrm{~m} / \mathrm{s}\). The second bird, with a mass of \(123 \mathrm{~g}\), is flying \(2.00^{\circ}\) east of north at a speed of \(11.0 \mathrm{~m} / \mathrm{s}\). The third bird, with a mass of \(112 \mathrm{~g}\), is flying \(22.0^{\circ}\) west of north at a speed of \(10.0 \mathrm{~m} / \mathrm{s}\). What is the momentum vector of the formation? What would be the speed and direction of a \(115-\mathrm{g}\) bird with the same momentum?

Short answer

Based on the given solution steps, create a short answer question: Question: Find the speed and direction of a 115 g bird that would have the same momentum as the formation of other birds with the following information: Bird 1: mass 100 g, velocity 8.00 m/s at 35° north of east. Bird 2: mass 123 g, velocity 11.0 m/s at 2° north of east. Bird 3: mass 112 g, velocity 10.0 m/s at 22° south of east. Answer: Following the given solution steps, the speed and direction of the 115 g bird with the same momentum as the formation can be found. The speed (v) would be the total momentum magnitude divided by the bird's mass, while the direction (θ) would be calculated using the tangent function for the angle. The final speed and direction will be the bird's flight characteristics.

Step-by-step solution

Find the momentum of each bird

To find the momentum of each bird, we need to multiply their mass by their velocity. First, we need to find the velocity components of each bird in x and y directions. Let's convert the mass of each bird to kg. Bird 1: \(100g = 0.1 kg\) Bird 2: \(123g = 0.123 kg\) Bird 3: \(112g = 0.112 kg\) For the x and y components of their velocity vectors: Bird 1: \(v_{1x} = 8.00\cos(35) ~m/s ,\,v_{1y} = 8.00\sin(35) ~m/s\) Bird 2: \(v_{2x} = 11.0\cos(2) ~m/s ,\,v_{2y} = 11.0\sin(2) ~m/s\) Bird 3: \(v_{3x} = -10.0\cos(22) ~m/s ,\,v_{3y} = 10.0\sin(22) ~m/s\) Now, we will find their respective momentum vector components. Bird 1: \(p_{1x} = 0.1v_{1x} ,\, p_{1y} = 0.1v_{1y}\) Bird 2: \(p_{2x} = 0.123v_{2x} ,\,p_{2y} = 0.123v_{2y}\) Bird 3: \(p_{3x} = 0.112v_{3x} ,\,p_{3y} = 0.112v_{3y}\)

Calculate the total momentum vector of the formation

To find the total momentum vector, we add up the momentum components in the x and y directions. \(p_{totalx} = p_{1x}+p_{2x}+p_{3x}\) \(p_{totaly} = p_{1y}+p_{2y}+p_{3y}\) Now, we obtain the total momentum vector. \(\vec{p}_{total} = p_{totalx} \hat{i} + p_{totaly} \hat{j}\)

Calculate the speed and direction of the 115 g bird with the same momentum

First, let's convert the mass of the bird to kg: \(115g = 0.115 kg\) Now, we will find the speed \(v\) and direction \(\theta\) of the bird with the same momentum vector as the formation. The magnitude of the momentum vector is: \(|\vec{p}_{total}| = \sqrt{p_{totalx}^2 + p_{totaly}^2}\) The speed of the bird is: \(v = \frac{|\vec{p}_{total}|}{m} = \frac{|\vec{p}_{total}|}{0.115}\) We can find the direction using the tangent function: \(\tan(\theta) = \frac{p_{totaly}}{p_{totalx}}\) To get the angle θ: \(\theta = \arctan\left(\frac{p_{totaly}}{p_{totalx}}\right)\)

Key concepts

Momentum in Physics

In the world of physics, momentum is a fundamental concept that describes the quantity of motion an object has. It's a vector quantity, meaning it has both magnitude and direction, and is defined by the product of an object's mass and velocity. This is captured in the equation: \[\begin{equation} momentum = mass \times velocity \(p = mv) \[5pt] \[5pt]\end{equation}\]For a complete understanding of momentum, it is essential to grasp that it's a conserved quantity in a closed system without external forces. In the exercise provided, we are examining the momentum of birds in flight, each with a distinct mass and velocity, giving rise to their individual momentum vectors. By calculating the total momentum of the formation, we understand how these vectors combine to describe the movement of the system as a whole.

Moreover, the conservation of momentum plays a critical role in different phenomena, from collisions in particle physics to understanding the motion of celestial bodies in astrophysics.

Vector Addition

Vector addition is pivotal in physics as it enables us to combine several vectors into a single resultant vector. It's akin to adding together multiple forces or motions that are acting on or within a system. In the context of the exercise, we calculate the momentum of the flying formation by adding up the momentum vectors of each bird. This process of vector addition follows specific rules, ensuring that both magnitude and direction of the vectors are considered.

To perform vector addition, one common method is the head-to-tail method, where the tail of each successive vector is placed at the head of the previous vector. Alternatively, you can also break down vectors into their horizontal (x) and vertical (y) components and then add these components separately:\[ \mathbf{R_x} = \sum\mathbf{V_{ix}} \]\[ \mathbf{R_y} = \sum\mathbf{V_{iy}} \]where \(\mathbf{R_x}\) and \(\mathbf{R_y}\) are the components of the resultant vector (R). These components are then used to find the magnitude and direction of the resultant.

Velocity Components

Velocity components are essential in understanding motion in more than one dimension. Every velocity vector can be decomposed into horizontal and vertical components, aligning with the x-axis and y-axis in a coordinate system. This decomposition simplifies the process of vector addition and other vector operations.

In our birds' flight formation scenario, we must consider the birds' velocities in terms of their respective eastward (or westward) and northward components, as given by:\[ v_x = v \times \cos(\theta) \]\[ v_y = v \times \sin(\theta) \]where \(v_x\) and \(v_y\) represent the horizontal and vertical components of the velocity, \(v\) is the speed, and \(\theta\) is the direction angle with respect to the northward direction. By converting the velocity into components, we are then able to apply these values to calculate the momentum in the respective directions, laying the groundwork to determine the total momentum of the formation.