Problem 65 - Fundamentals of Physics

Question

A protester carries his sign of protest, starting from the origin of an \(x y z\) coordinate system, with the \(x y\) plane horizontal. He moves \(40 \mathrm{~m}\) in the negative direction of the \(x\) axis, then \(20 \mathrm{~m}\) along a perpendicular path to his left, and then \(25 \mathrm{~m}\) up a water tower. (a) In unit-vector notation, what is the displacement of the sign from start to end? (b) The sign then falls to the foot of the tower. What is the magnitude of the displacement of the sign from start to this new end?

Step-by-Step Solution

Answer

(a) \(-40\hat{i} - 20\hat{j} + 25\hat{k}\). (b) Magnitude: \(\sqrt{2000}\).

Step 1: Identifying Initial Displacement

The protester starts at the origin (0,0,0). Moving 40 m in the negative x-direction means the displacement in terms of vectors is \(-40\hat{i}\).

Step 2: Identifying Perpendicular Displacement

The protester then moves 20 m along a path perpendicular and to the left of the x-axis. In our coordinate system, this means moving in the y-direction. Hence, the displacement here is \(-20\hat{j}\) because it is to the left, aligning with the negative y-direction.

Step 3: Identifying Vertical Displacement

Finally, the protester climbs 25 m up the water tower, which is a movement in the positive z-axis. Therefore, this displacement is represented as \(+25\hat{k}\).

Step 4: Calculating Total Displacement

Adding all components together, the total displacement in unit-vector notation is:\[\vec{d} = -40\hat{i} - 20\hat{j} + 25\hat{k} \].

Step 5: Calculating Magnitude before Falling

To find the magnitude of this displacement, we use the formula for the magnitude of a vector:\[|\vec{d}| = \sqrt{(-40)^2 + (-20)^2 + (25)^2}\]. Evaluating this:\[|\vec{d}| = \sqrt{1600 + 400 + 625} = \sqrt{2625}\].

Step 6: Calculating Displacement After Falling

When the sign falls back to the foot of the tower, the z-component becomes zero again. Thus, the new displacement vector is:\(-40\hat{i} - 20\hat{j}\), so the magnitude is:\[|\vec{d}| = \sqrt{(-40)^2 + (-20)^2} = \sqrt{1600 + 400} = \sqrt{2000}\].

Key Concepts

Unit-Vector Notation

When describing displacement in a three-dimensional space like the one used in this exercise, unit-vector notation is an efficient way to detail the direction and magnitude of movement along each coordinate axis. Each unit vector serves to specify one direction:

• \(\hat{i}\) indicates movement along or opposite to the x-axis,
• \(\hat{j}\) is used for the y-axis,
• \(\hat{k}\) represents the z-axis.

Thus, expressing displacement as a combination of these vectors shows how an object moves through each component axis. For the protester moving through different axes, we write the displacement as \(-40\hat{i} - 20\hat{j} + 25\hat{k}\), summarizing his journey's path from the start to the end point.

Magnitude of Displacement

The magnitude of displacement refers to the straight-line distance between the starting and ending points of a movement, regardless of the path taken. This value gives us an idea of the actual distance covered as the crow flies.
In mathematical terms, this is calculated using the Pythagorean theorem extended to three dimensions: \[|\vec{d}| = \sqrt{(x_1 - x_0)^2 + (y_1 - y_0)^2 + (z_1 - z_0)^2}\]Where \((x_0, y_0, z_0)\) and \((x_1, y_1, z_1)\) are the initial and final positions, respectively. In our example, the magnitude before the sign falls is calculated as \(|\vec{d}| = \sqrt{2625}\), giving a solid measure of the journey's overall displacement.

Coordinate System

A coordinate system helps us define positions in space with numerical coordinates. In this exercise, we are dealing with a three-dimensional Cartesian coordinate system where:

• the x-axis runs horizontally,
• the y-axis also runs horizontally but perpendicular to the x-axis,
• the z-axis runs vertically.

Each point in this 3D space is defined by an ordered triplet \((x, y, z)\). Starting at the origin \((0, 0, 0)\), each movement is tracked along these axes, making it clear how far the protester has moved in each direction. This system allows us to clearly visualize and calculate the protester's movements.

Displacement Components

To understand a displacement fully in three-dimensional space, it is critical to break it down into components along each axis of the coordinate system. This method provides a clearer view of how displacement happens in different directions.
For example:

• The x-component was \(-40\hat{i}\), resulting from moving 40 meters in the negative direction of the x-axis.
• The y-component \(-20\hat{j}\) comes from moving 20 meters perpendicular to the x-axis.
• The z-component \(+25\hat{k}\) results from ascending 25 meters upwards along the tower.

By summing these components, we understand the total displacement vector \(-40\hat{i} - 20\hat{j} + 25\hat{k}\), illustrating the cumulative effect of each segment of the protester's path.