Problem 28 - College Physics

Question

The velocity of object 1 in component form is \(\overrightarrow{\mathbf{v}}_{1}=(+2.0 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{x}}+(-4.0 \mathrm{~m} / \mathrm{s}) \hat{\mathbf{y}} .\) Object 2 has twice the speed of object 1 but moves in the opposite direction. (a) Determine the velocity of object 2 in component notation. (b) What is the speed of object \(2 ?\)

Step-by-Step Solution

Answer

(a) \(\overrightarrow{v}_2 = (-4.0 \, \text{m/s}) \hat{\mathbf{x}} + (+8.0 \, \text{m/s}) \hat{\mathbf{y}}\); (b) 8.94 m/s.

Step 1: Determine the Speed of Object 1

First, calculate the speed of object 1 using its velocity components. The speed is the magnitude of the velocity vector. The formula to determine this magnitude is: \[ \text{Speed} = \sqrt{v_x^2 + v_y^2} \]Substitute the given components: \[ \text{Speed of Object 1} = \sqrt{(2.0)^2 + (-4.0)^2} \]\[ = \sqrt{4.0 + 16.0} = \sqrt{20.0} = \sqrt{4.0 \times 5.0} = 2.0 \times \sqrt{5.0} \approx 4.47 \, \text{m/s} \]

Step 2: Calculate the Speed of Object 2

Object 2 has twice the speed of Object 1. So we take the speed from Step 1 and multiply it by 2: \[ \text{Speed of Object 2} = 2 \times 4.47 \, \text{m/s} = 8.94 \, \text{m/s} \]

Step 3: Determine the Components of Object 2's Velocity

Since Object 2 moves in the opposite direction, each component of velocity will be the negative of Object 1's, scaled by the speed ratio, which is 2.- For the x-component: \[ v_{2x} = -2 \times (+2.0) = -4.0 \, \text{m/s} \]- For the y-component:\[ v_{2y} = -2 \times (-4.0) = +8.0 \, \text{m/s} \]Thus, the velocity of Object 2 in component form is:\[ \overrightarrow{v}_2 = (-4.0 \, \text{m/s}) \hat{\mathbf{x}} + (+8.0 \, \text{m/s}) \hat{\mathbf{y}} \]

Key Concepts

Velocity Calculation

Velocity is a crucial concept in physics as it describes both the speed and direction of an object. When calculating velocity, particularly in problems involving components, it’s important to pay attention to both the magnitude and the direction indicated by the positive or negative signs of the components.

• Velocity is a vector quantity, meaning it has both magnitude and direction.
• In component form, velocity is expressed as separate horizontal and vertical components. For example, \( \overrightarrow{\mathbf{v}}_{1}=(+2.0 \mathrm{~m/s}) \hat{\mathrm{x}}+(-4.0 \mathrm{~m/s}) \hat{\mathbf{y}} \).
• The magnitude of the velocity provides the speed of the object, while the negative or positive signs indicate direction along the respective axes.

To calculate the velocity components for object 2, we not only double the speed, but we also reverse the direction of each component. This gives us a new velocity formula which has both a magnitude that is double that of object 1 and an opposite direction.

Vector Components

Understanding vector components is key to solving physics problems involving direction and magnitude. A vector in two-dimensional space can be broken down into two parts: a horizontal component and a vertical component.

• The x-component represents the horizontal movement of the vector. It is aligned along the x-axis.
• The y-component represents the vertical movement of the vector, aligned along the y-axis.

When vectors change direction, as in the exercise with object 2, their components must reflect this change. Object 1 has these components: \[ +(2.0 \mathrm{~m/s}) \hat{\mathrm{x}} \text{ and } (-4.0 \mathrm{~m/s}) \hat{\mathbf{y}} \].For object 2 moving in the opposite direction and at twice the speed, each component becomes negative and scaled: \[ v_{2x} = -4.0 \mathrm{~m/s} \] and \[ v_{2y} = +8.0 \mathrm{~m/s} \].These adjusted components determine the overall direction of object 2.

Speed Determination

Speed, a scalar quantity, differs from velocity only in that it does not have direction—only magnitude. When calculating speed from velocity components, you focus on these components' absolute values.To determine the speed of an object using its velocity components, calculate the square root of the sum of the squares of each component:\[\text{Speed} = \sqrt{v_x^2 + v_y^2}\]For object 1, this computes to:\[ \sqrt{(2.0)^2 + (-4.0)^2} \approx 4.47 \, \text{m/s} \].Object 2, moving at twice this speed, has:\[ 2 \times 4.47 = 8.94 \, \text{m/s} \].The critical takeaway is that the calculation of speed helps deduce the overall magnitude without considering the direction of motion.