Question

You travel from the location \(x_{\mathrm{i}}=20 \mathrm{~m}\) to the location \(x_{\mathrm{f}}=25 \mathrm{~m}\). Your friend travels from \(x_{\mathrm{i}}=35 \mathrm{~m}\) to \(x_{\mathrm{f}}=30 \mathrm{~m}\). Which of you has a positive displacement? Which of you has a negative displacement? Explain.

Short answer

You have a positive displacement (+5 m); your friend has a negative displacement (-5 m).

Step-by-step solution

Understanding Displacement

Displacement is the change in position of an object. It is calculated as the final position minus the initial position. If the result is positive, the object has moved in the positive direction; if negative, in the negative direction.

Calculating Your Displacement

For your journey, the initial position is given as \( x_i = 20 \, \text{m} \) and the final position is \( x_f = 25 \, \text{m} \). The displacement \( \Delta x \) is calculated by \( \Delta x = x_f - x_i = 25 - 20 = 5 \, \text{m} \). Since the displacement is positive, you have a positive displacement.

Calculating Your Friend's Displacement

For your friend's journey, the initial position is \( x_i = 35 \, \text{m} \) and the final position is \( x_f = 30 \, \text{m} \). The displacement \( \Delta x \) is \( \Delta x = x_f - x_i = 30 - 35 = -5 \, \text{m} \). Since the displacement is negative, your friend has a negative displacement.

Comparing Displacements

You moved from 20 m to 25 m, resulting in a positive displacement of +5 m, whereas your friend moved from 35 m to 30 m, resulting in a negative displacement of -5 m. This means you have a positive displacement and your friend has a negative displacement.

Key concepts

Positive Displacement

When assessing positive displacement, it's crucial to understand that it indicates movement in the intended or designated direction. In the context of physics, this generally means an increase in position value along a defined path. For example, if you start at 20 meters and travel to 25 meters, your displacement is \( \Delta x = x_f - x_i = 25 - 20 = 5 \, \text{m} \) resulting in a positive displacement of 5 meters.

• Positive displacement means you've moved forward in your frame of reference.
• This concept is key in understanding motion as it relates to directionality.

Displacement tells us not only how far you've moved but also in which direction relative to your starting point.

Negative Displacement

Negative displacement occurs when your position changes in the opposite direction to the one typically considered positive. It implies that you have moved backward along a pathway. Let's say your friend starts at 35 meters and ends at 30 meters. The calculation looks like this:\( \Delta x = x_f - x_i = 30 - 35 = -5 \, \text{m} \).That results in a negative displacement of -5 meters.

• Negative displacement shows movement in the reverse direction from the expected norm.
• This helps to identify shifts that result in a reduction of position value.

Remember, displacement takes into account both magnitude and direction, which is why the sign (positive or negative) is significant.

Calculating Displacement

Calculating displacement involves a simple yet powerful formula: subtract the initial position from the final position:\( \Delta x = x_f - x_i \).It's crucial to correctly identify these positions for accurate calculation. Think of displacement as the shortest path between two points, marked by a line from where you began to where you finished. This formula works in various contexts:

• A positive result indicates advancement or progression.
• A negative result signifies regression or movement in the opposite direction.

Displacement is a vector quantity, meaning it accounts for both direction and size, differentiating it from distance, which lacks a directional component. This makes displacement invaluable in fields like physics and engineering, where understanding the direction of movement is just as important as knowing how far something has traveled. Temperature management, efficient resource use, and even robotic navigation all rely on precise displacement calculations.