Examples on Motion of a Subway Train
The next several examples consider the motion of the subway train shown in the figure below. In (a) the shuttle moves to the right, and in (b) it moves to the left. The examples are designed to further illustrate aspects of motion and to illustrate some of the reasoning that goes into solving problems.
One-dimensional motion of a subway train considered in the examples below. Here we have chosen the x-axis so that + means to the right and − means to the left for displacements, velocities, and accelerations. (a) The subway train moves to the right from x0 to xf. Its displacement Δx is +2.0 km. (b) The train moves to the left from x′0 to x′f. Its displacement Δx′ is −1.5 km. (Note that the prime symbol (′) is used simply to distinguish between displacement in the two different situations. The distances of travel and the size of the cars are on different scales to fit everything into the diagram.) Image credit: OpenStax Physics
Example on Calculating Displacement: A Subway Train
What are the magnitude and sign of displacements for the motions of the subway train shown in parts (a) and (b) of the figure above?
Strategy
A drawing with a coordinate system is already provided, so we don’t need to make a sketch, but we should analyze it to make sure we understand what it is showing. Pay particular attention to the coordinate system. To find displacement, we use the equation \(\Delta x = x_{\text{f}} - x_0.\) This is straightforward since the initial and final positions are given.
Solution
1. Identify the knowns. In the figure we see that \(x_{\text{f}} = 6.70\text{ km}\) and \(x_0 = 4.70\text{ km}\) for part (a), and \(x^{\prime}_{\text{f}} = 3.75\text{ km}\) and \(x^{\prime}_0 = 5.25\text{ km}\) for part (b).
2. Solve for displacement in part (a).
\(\Delta x = x_{\text{f}} \; - \; x_0 = 6.70\text{ km} \; - \; 4.70\text{ km} = +2.00\text{ km}\)
3. Solve for displacement in part (b).
\(\Delta x' = x'_{\text{f}} \; - \; x'_0 = 3.75\text{ km} \; - \; 5.25\text{ km} = -1.50\text{ km}\)
Discussion
The direction of the motion in (a) is to the right and therefore its displacement has a positive sign, whereas motion in (b) is to the left and thus has a negative sign.
Example on Comparing Distance Traveled with Displacement: A Subway Train
What are the distances traveled for the motions shown in parts (a) and (b) of the subway train in the figure above?
Strategy
To answer this question, think about the definitions of distance and distance traveled, and how they are related to displacement. Distance between two positions is defined to be the magnitude of displacement, which was found in the example above on calculating displacement.
Distance traveled is the total length of the path traveled between the two positions. (See the lesson on displacement.) In the case of the subway train shown in the figure above, the distance traveled is the same as the distance between the initial and final positions of the train.
Solution
1. The displacement for part (a) was +2.00 km. Therefore, the distance between the initial and final positions was 2.00 km, and the distance traveled was 2.00 km.
2. The displacement for part (b) was −1.5 km.
Therefore, the distance between the initial and final positions was 1.50 km, and the distance traveled was 1.50 km.
Discussion
Distance is a scalar. It has magnitude but no sign to indicate direction.
This lesson is part of:
One-Dimensional Kinematics