Question

find the following for the path in the figure below.
(a) the total distance traveled
m
(b) the displacement from start to finish
m
resources
reading

Explanation:

Part (a): Total Distance Traveled

Step1: Analyze the Path Segments

The start point is at \( x = 2 \, \text{m} \) (from the origin, as the first segment starts at the red dot near \( x = 2 \)). The main horizontal segment goes from \( x = 2 \) to \( x = 10 \), then there's a small back-and-forth (a "U - turn" like shape). Let's break down the path:

• From start (\( x = 2 \)) to the start of the U - turn: distance is \( 10 - 2=8 \, \text{m} \).
• The U - turn: Let's assume the U - turn has two small segments. Looking at the graph, the U - turn goes from \( x = 8 \) to \( x = 10 \) (back) and then to \( x = 8 \) (forward)? Wait, no, the figure shows a path that goes straight to a point, then does a small loop (two small horizontal segments: one back, one forward) and then continues. Wait, actually, the start is at \( x = 2 \), then it goes to \( x = 10 \), then back to \( x = 8 \), then forward to \( x = 10 \), then forward to \( x = 12 \)? Wait, no, the x - axis has marks at 0, 2, 4, 6, 8, 10, 12. The start point is at \( x = 2 \). The path goes from \( x = 2 \) to \( x = 10 \) (distance \( 10 - 2 = 8 \, \text{m} \)), then back to \( x = 8 \) (distance \( 10 - 8=2 \, \text{m} \)), then forward to \( x = 10 \) (distance \( 10 - 8 = 2 \, \text{m} \)), then forward to \( x = 12 \) (distance \( 12 - 10=2 \, \text{m} \))? Wait, no, maybe a better way: the total distance is the sum of all the lengths of the path. The straight part from \( x = 2 \) to \( x = 12 \) would be \( 12 - 2 = 10 \, \text{m} \), but with the two small back - and - forth segments (each of length \( 2 \, \text{m} \), since from \( x = 8 \) to \( x = 10 \) is \( 2 \, \text{m} \), and back to \( x = 8 \) is another \( 2 \, \text{m} \), then forward to \( x = 10 \) and then to \( x = 12 \)? Wait, no, looking at the figure, the path has a start at \( x = 2 \), then goes to \( x = 10 \), then does a loop: goes back \( 2 \, \text{m} \) (to \( x = 8 \)), then forward \( 2 \, \text{m} \) (to \( x = 10 \)), then forward \( 2 \, \text{m} \) (to \( x = 12 \)). Wait, no, the correct way: the total distance is the length of the entire path. The start is at \( x = 2 \), the end is at \( x = 12 \). But the path has a detour: from \( x = 8 \) to \( x = 10 \) (forward), then back to \( x = 8 \) (backward), then forward to \( x = 10 \) and then to \( x = 12 \). Wait, actually, the horizontal distance from start (\( x = 2 \)) to the point before the loop is \( 10 - 2=8 \, \text{m} \). Then the loop: from \( x = 8 \) to \( x = 10 \) (distance \( 2 \, \text{m} \)), back to \( x = 8 \) (distance \( 2 \, \text{m} \)), then from \( x = 8 \) to \( x = 12 \) (distance \( 4 \, \text{m} \))? No, that can't be. Wait, maybe the loop is two segments: each of length \( 2 \, \text{m} \) (so total loop distance \( 2 + 2=4 \, \text{m} \)), and the straight part from \( x = 2 \) to \( x = 12 \) is \( 10 \, \text{m} \), but with the loop, the total distance is \( (12 - 2)+4=14 \, \text{m} \)? Wait, let's think again. The start is at \( x = 2 \). The path goes:
• From \( x = 2 \) to \( x = 10 \): distance \( 10 - 2 = 8 \, \text{m} \)
• Then from \( x = 10 \) to \( x = 8 \): distance \( 10 - 8 = 2 \, \text{m} \)
• Then from \( x = 8 \) to \( x = 10 \): distance \( 10 - 8 = 2 \, \text{m} \)
• Then from \( x = 10 \) to \( x = 12 \): distance \( 12 - 10 = 2 \, \text{m} \)

Now sum these up: \( 8+2 + 2+2=14 \, \text{m} \)

Step2: Sum the Distances

Add the lengths of each segment: \( 8+2 + 2+2 = 14 \, \text{m} \)

Part (b): Displacement from Start to Finish

Step1: Define Displacement

Displacement is the straight - line distance from the initial position to the final position, along with the direction (in one - dimension, it's the difference between final and initial position).

Step2: Calculate Displacement

Initial position \( x_i=2 \, \text{m} \), final position \( x_f = 12 \, \text{m} \). Displacement \( \Delta x=x_f - x_i=12 - 2 = 10 \, \text{m} \)

Answer:

(a):
\( 14 \)