Problem 85 - University Physics with Modern Physics

Question

Three birds are flying in a compact formation. The first bird, with a mass of \(100 . \mathrm{g}\) 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?

Step-by-Step Solution

Answer

Answer: The speed and direction of the fourth bird are given by the formula velocity4 and direction4, which are calculated by dividing the magnitude of the total momentum vector by the mass of the fourth bird and using the direction of the total momentum vector, respectively.

Step 1: Calculate the momentum of each bird individually

First, we'll find the momentum of each bird in the formation using the formula: momentum = mass × velocity. Since each bird has different mass, velocity, and direction, we'll find the respective momentum of each bird along the x and y axis, respectively.

Step 1.1: Calculate the x and y components of the momentum for the first bird

The mass of the first bird is \(100\) g. Convert this mass into kg by dividing by 1000: Mass \(= \frac{100}{1000} = 0.1 \,\text{kg}\) Now, find the x and y components of its velocity using trigonometry: v_x1 = v1 * cos(35.0°) v_y1 = v1 * sin(35.0°) Calculate the momentum in the x and y directions: momentum_x1 = mass1 * v_x1 momentum_y1 = mass1 * v_y1

Step 1.2: Calculate the x and y components of the momentum for the second bird

The mass of the second bird is \(123\) g. Convert this mass into kg by dividing by 1000: Mass \(= \frac{123}{1000} = 0.123 \,\text{kg}\) Now, find the x and y components of its velocity using trigonometry: v_x2 = v2 * cos(2.0°) v_y2 = v2 * sin(2.0°) Calculate the momentum in the x and y directions: momentum_x2 = mass2 * v_x2 momentum_y2 = mass2 * v_y2

Step 1.3: Calculate the x and y components of the momentum for the third bird

The mass of the third bird is \(112\) g. Convert this mass into kg by dividing by 1000: Mass \(= \frac{112}{1000} = 0.112 \,\text{kg}\) Now, find the x and y components of its velocity using trigonometry: v_x3 = v3 * cos(-22.0°) v_y3 = v3 * sin(-22.0°) Calculate the momentum in the x and y directions: momentum_x3 = mass3 * v_x3 momentum_y3 = mass3 * v_y3

Step 2: Calculate the momentum vector of the formation

Now that we have the momentum components of each bird in the x and y directions, we can find the total momentum vector of the formation by summing the individual components: total_momentum_x = momentum_x1 + momentum_x2 + momentum_x3 total_momentum_y = momentum_y1 + momentum_y2 + momentum_y3 Now that we have the total momentum in x and y directions, we can find the magnitude and direction of the momentum vector using Pythagorean theorem and arctan function: magnitude = sqrt(total_momentum_x^2 + total_momentum_y^2) direction = atan(total_momentum_y / total_momentum_x)

Step 3: Calculate the speed and direction of the fourth bird

Given that the fourth bird has a mass of \(115\) g and the same momentum as the formation, we can find its speed and direction. First, convert the mass into kg: mass4 = \(\frac{115}{1000} = 0.115 \,\text{kg}\) Now, use the momentum magnitude we found in step 2 to find the fourth bird's velocity: velocity4 = magnitude / mass4 Use the direction of the momentum vector to find the direction in which the fourth bird should fly: direction4 = direction Now we have the speed and direction of the fourth bird that has the same momentum as the formation.

Key Concepts

Physics Momentum

When we talk about momentum in physics, we're discussing a measurement of the quantity of motion an object possesses. This concept is vital in understanding both motion and collisions. The momentum of an object is determined by two factors: its mass and its velocity. In mathematical terms, momentum (\textbf{p}) is expressed as the product of an object's mass (m) and its velocity (\textbf{v}), so the formula is \[ \textbf{p} = m \times \textbf{v} \].

In a scenario with multiple objects, like our birds in formation, the total momentum of the system is the vector sum of all individual momenta. To compute this in a problem involving different directions, we need to consider the components of the momentum vector.

In the exercise provided, as each bird flies at a different speed and direction, their individual momenta contribute uniquely to the formation's overall momentum. This accumulation is essential in determining the behavior of the formation if any external forces act upon it. Moreover, in a closed system, the law of conservation of momentum states that the total momentum remains constant, unless acted upon by an external force.

Vector Components

Vector components are a way to break down a vector into its parts along the axes of the coordinate system, usually along the x (horizontal) and y (vertical) axes. This breakdown is immensely helpful in simplifying calculations and understanding the motion in each direction independently.

For instance, calculating the momentum of each bird means splitting their velocities into horizontal and vertical parts. This is done using trigonometry, specifically the sine and cosine functions, which relate the angle of motion to the respective x and y components.

Once we have calculated the x (horizontal) and y (vertical) components of each bird's momentum, we can find the total momentum for the formation by adding up these components independently. The formula for a momentum component along an axis is \[ \text{momentum}_{\text{axis}} = \text{mass} \times \text{velocity}_{\text{axis}} \].

In the birds' formation, we calculate these components by first converting mass into kilograms and then applying the cosine (for the x-axis) and sine (for the y-axis) to the birds' velocities in conjunction with their flight angles. This operation is key to understanding vector quantities' roles in various physical situations and plays an integral part in problem-solving.

Trigonometry in Physics

Trigonometry is not just a topic in mathematics but also an essential tool in physics, particularly when dealing with vector quantities like velocity, force, and momentum. In our exercise, trigonometry helps determine the direction and magnitude components of vectors.

The two trigonometric functions that come into play with vectors are sine (sin) and cosine (cos). They are used to calculate the x and y components of a vector, given its magnitude and angle of direction. For example, if a bird is flying at an angle \( \theta \) from a reference direction, the velocity components can be found using \[ \text{velocity}_x = \text{velocity} \times \cos(\theta) \] and \[ \text{velocity}_y = \text{velocity} \times \sin(\theta) \].

In the context of our problem, this trigonometric approach allows us to deconstruct each bird's velocity vector into more manageable x and y components, which can be used to find the overall momentum of the bird formation. It's this combination of trigonometry and vector analysis that proves invaluable for solving a wide array of physics problems involving direction and magnitude.