Problem 3 - Fundamentals of Physics

Question

A positron undergoes a displacement \(\Delta \vec{r}=2.0 \hat{\mathrm{i}}-3.0 \hat{\mathrm{j}}+6.0 \hat{\mathrm{k}}\), ending with the position vector \(\vec{r}=3.0 \hat{\mathrm{j}}-4.0 \hat{\mathrm{k}}\), in meters. What was the positron's initial position vector? w.4 The minute hand of a wall clock measures \(10 \mathrm{~cm}\) from its tip to the axis about which it rotates. The magnitude and angle of the displacement vector of the tip are to be determined for three time intervals. What are the (a) magnitude and (b) angle from a quarter after the hour to half past, the (c) magnitude and (d) angle for the next half hour, and the (e) magnitude and (f) angle for the hour after that?

Step-by-Step Solution

Answer

The initial position vector is \( \vec{r}_0 = -2.0 \hat{\mathrm{i}} + 6.0 \hat{\mathrm{j}} - 10.0 \hat{\mathrm{k}} \).

Step 1: Identify the Final Position Vector Components

The final position vector of the positron is given as \( \vec{r} = 3.0 \hat{\mathrm{j}} - 4.0 \hat{\mathrm{k}} \). This means the components of the final position vector are \( x_f = 0 \), \( y_f = 3.0 \), and \( z_f = -4.0 \).

Step 2: Identify the Displacement Vector Components

The displacement vector \( \Delta \vec{r} \) is given as \( \Delta \vec{r} = 2.0 \hat{\mathrm{i}} - 3.0 \hat{\mathrm{j}} + 6.0 \hat{\mathrm{k}} \). This means the components of the displacement vector are \( \Delta x = 2.0 \), \( \Delta y = -3.0 \), and \( \Delta z = 6.0 \).

Step 3: Apply the Displacement Formula

To find the initial position vector \( \vec{r}_0 \), we use the equation \( \vec{r} = \vec{r}_0 + \Delta \vec{r} \). This can be rearranged to find \( \vec{r}_0 = \vec{r} - \Delta \vec{r} \). Substitute the final position vector and displacement vector components into the equation.

Step 4: Calculate the Initial Position Vector Components

Calculate each component of the initial position vector:\[ x_0 = x_f - \Delta x = 0 - 2.0 = -2.0 \]\[ y_0 = y_f - \Delta y = 3.0 - (-3.0) = 6.0 \]\[ z_0 = z_f - \Delta z = -4.0 - 6.0 = -10.0 \]

Step 5: Write the Initial Position Vector

The initial position vector \( \vec{r}_0 \) can now be written as:\[ \vec{r}_0 = -2.0 \hat{\mathrm{i}} + 6.0 \hat{\mathrm{j}} - 10.0 \hat{\mathrm{k}} \]

Key Concepts

Initial Position Vector

In physics, understanding the initial position vector is crucial in describing the motion of an object. The initial position vector, usually denoted as \( \vec{r}_0 \), represents the starting point of an object in three-dimensional space. It is typically expressed in terms of its components in the x, y, and z directions: \( \vec{r}_0 = x_0\hat{\mathrm{i}} + y_0\hat{\mathrm{j}} + z_0\hat{\mathrm{k}} \).

When dealing with problems involving motion or displacement, knowing the initial position vector allows us to track how much an object has moved from its starting point. In the exercise, the positron's initial position vector is determined by subtracting the displacement vector from its final position vector. Each component of the position vector provides detailed information about the object's initial position in each of the three axes:

• \( x_0 = x_f - \Delta x \)
• \( y_0 = y_f - \Delta y \)
• \( z_0 = z_f - \Delta z \)

This calculated initial position vector gives a comprehensive snapshot of where the positron began its journey in space.

Final Position Vector

The final position vector is a key concept in vector displacement problems, indicating the object's position after undergoing a movement. Notated as \( \vec{r} \), it provides the coordinates of an object at the conclusion of its motion. For three-dimensional spaces, the vector is expressed as \( \vec{r} = x_f\hat{\mathrm{i}} + y_f\hat{\mathrm{j}} + z_f\hat{\mathrm{k}} \).

Understanding the final position vector is essential for both tracking movement and calculating other vectors such as displacement or initial position. In the problem, the final position vector of the positron is given with components: \( x_f = 0 \), \( y_f = 3.0 \), and \( z_f = -4.0 \). These values reflect the ending location along each axis after the motion has occurred.

To find the initial position vector using the given final position, we incorporate the displacement vector. The relationship and computations between these vectors highlight how the object moved and set the stage for analyzing its path through space.

Displacement Vector Components

The displacement vector is a fundamental vector concept that describes how an object's position changes over time. It is represented as \( \Delta \vec{r} = \Delta x\hat{\mathrm{i}} + \Delta y\hat{\mathrm{j}} + \Delta z\hat{\mathrm{k}} \), and is found by subtracting the initial position vector from the final position vector.

In the given exercise, the displacement vector signifies the movement of the positron through space, with the components being \( \Delta x = 2.0 \), \( \Delta y = -3.0 \), and \( \Delta z = 6.0 \). Each part of this vector tells us how far and in what direction the positron has shifted along those axes.

• The positive \( \Delta x \) indicates a move rightward along the x-axis.
• The negative \( \Delta y \) shows a downward movement along the y-axis.
• The positive \( \Delta z \) points to an upward movement along the z-axis.

This vector plays a pivotal role in both determining the object's initial position and understanding its trajectory during the movement. It connects the initial and final positions, allowing us to back-calculate and interpret the positron's journey in a clear, component-based manner.