Problem 4 - College Physics

Question

\(\cdot\) A coyote chasing a rabbit is moving 8.00 \(\mathrm{m} / \mathrm{s}\) due east at one moment and 8.80 \(\mathrm{m} / \mathrm{s}\) due south 4.00 s later. Find (a) the \(x\) and \(y\) components of the coyote's average acceleration during that time and (b) the magnitude and direction of the coyote's average acceleration during that time.

Step-by-Step Solution

Answer

The average acceleration is \(2.97\, \text{m/s}^2\) at \(47.7^{\circ}\) south of east.

Step 1: Understanding the Initial and Final Velocities

The coyote starts moving at 8.00 m/s due east. This can be considered as the initial velocity vector \(\mathbf{v}_i = 8.00\, \text{m/s}\) in the positive x-direction, which means \(v_{ix} = 8.00\, \text{m/s}\) and \(v_{iy} = 0\, \text{m/s}\). After 4 seconds, the coyote is moving 8.80 m/s due south. This final velocity vector \(\mathbf{v}_f\) points entirely along the negative y-direction, so \(v_{fx} = 0\, \text{m/s}\) and \(v_{fy} = -8.80\, \text{m/s}\).

Step 2: Calculating the Change in Velocity Components

Determine the change in each component of velocity. For the x-component, it changes from \(v_{ix} = 8.00\, \text{m/s}\) to \(v_{fx} = 0\, \text{m/s}\). Thus, the change in x-component is \(\Delta v_x = v_{fx} - v_{ix} = 0 - 8.00 = -8.00\, \text{m/s}\). For the y-component, it changes from \(v_{iy} = 0\, \text{m/s}\) to \(v_{fy} = -8.80\, \text{m/s}\). Thus, the change in y-component is \(\Delta v_y = v_{fy} - v_{iy} = -8.80 - 0 = -8.80\, \text{m/s}\).

Step 3: Finding the Average Acceleration Components

Use the formula for average acceleration, \( a = \frac{\Delta v}{\Delta t} \). The average acceleration in the x-direction is \( a_x = \frac{\Delta v_x}{\Delta t} = \frac{-8.00\, \text{m/s}}{4.00\, \text{s}} = -2.00\, \text{m/s}^2 \). Similarly, the average acceleration in the y-direction is \( a_y = \frac{\Delta v_y}{\Delta t} = \frac{-8.80\, \text{m/s}}{4.00\, \text{s}} = -2.20\, \text{m/s}^2 \).

Step 4: Calculating the Magnitude of the Average Acceleration

The magnitude of the average acceleration vector \(\mathbf{a}\) can be found using the Pythagorean theorem: \( |\mathbf{a}| = \sqrt{a_x^2 + a_y^2} = \sqrt{(-2.00)^2 + (-2.20)^2} = \sqrt{4.00 + 4.84} = \sqrt{8.84} \approx 2.97\, \text{m/s}^2 \).

Step 5: Determining the Direction of the Average Acceleration

To find the direction, calculate the angle \(\theta\) relative to the negative x-axis using \( \tan \theta = \frac{a_y}{a_x} \). Thus, \( \theta = \tan^{-1}\left(\frac{-2.20}{-2.00}\right) = \tan^{-1}(1.10) \approx 47.7^{\circ} \). Since both components are negative, the direction is towards the southeast, specifically \(47.7^{\circ}\) south of east.

Key Concepts

Kinematics

Kinematics is a branch of physics that deals with motion without considering the forces that cause it. It involves describing the motion in terms of displacement, velocity, and acceleration. These parameters help in understanding how an object moves. In the coyote-chasing-a-rabbit problem, we explore the motion by focusing on how the velocity changes over time. The time interval provided (4 seconds) is crucial, as it allows us to calculate average acceleration, which gives a good idea of how quickly the coyote's speed and direction have changed. This concept enables us to determine how the coyote manages its chase, given the directional changes it undergoes.

Velocity Components

Velocity is a vector quantity, meaning it has both magnitude and direction. When an object, like the coyote in our exercise, changes direction, we break down its velocity into components. Velocity components help in understanding the motion in different directions—usually considering horizontal (x-axis) and vertical (y-axis) components. By initially traveling east and later turning south, the coyote's velocity has changed in both the x and y directions. We calculate these changes to determine the average acceleration components. The x-component of velocity decreased from 8.00 m/s to 0 m/s, while the y-component changed from 0 m/s to -8.80 m/s. Such breakdowns simplify otherwise complex changes in motion.

Pythagorean Theorem

The Pythagorean theorem is used to calculate the resultant vector when dealing with components of motion. In our problem, once we find the individual average acceleration components (-2.00 m/s² and -2.20 m/s² for x and y respectively), we need to determine the overall magnitude of the average acceleration. This is where the Pythagorean theorem comes handy. We apply it by taking the square root of the sum of the squares of these components: \[ a = \sqrt{a_x^2 + a_y^2} \] Applying this method, we find that the magnitude of the coyote's average acceleration is approximately 2.97 m/s². It offers a single value for understanding how fast the velocity is changing, regardless of direction.

Direction of Acceleration

Understanding the direction of average acceleration is as important as knowing its magnitude. The direction is calculated using the tangent function, which relates the ratio of the acceleration components. For the coyote, we find the angle \( \theta \) from the negative x-axis (since its acceleration direction is in the third quadrant). By using \( \tan \theta = \frac{a_y}{a_x} \), we calculate the angle to be approximately 47.7° south of east. Knowing this angle is vital, as it specifies in which direction the coyote is accelerating. This directional information, combined with magnitude, provides a complete picture of the coyote's motion during the chase.