Question

Rank Four trains travel on different sections of a long straight track. Taking north to be the positive direction, rank the trains in order by velocity, from most negative to most positive. The trains move as follows: Train A moves north with a speed of \(10 \mathrm{~m} / \mathrm{s}\). Train B heads south and covers \(900 \mathrm{~m}\) in \(1 \mathrm{~min}\). Train \(\mathrm{C}\) also heads south and has twice the speed of train \(A\). Train D travels north and covers \(24 \mathrm{~m}\) in \(2 \mathrm{~s}\).

Short answer

Rank: Train C, Train B, Train A, Train D.

Step-by-step solution

Determine direction and notation

First, understand that velocities moving south are negative, and velocities moving north are positive. We need to calculate velocities for trains that do not provide explicit information.

Calculate Train B's velocity

Train B covers \(900 \text{ m}\) in \(1 \text{ minute}\). \[ v_B = \frac{-900 \text{ m}}{60 \text{ s}} = -15 \text{ m/s}. \] The velocity is negative because the train is moving south.

Calculate Train C's velocity

Train C heads south with twice the speed of Train A. Since Train A moves north at 10 m/s, Train C moves south at \[ v_C = -2 \times 10 \text{ m/s} = -20 \text{ m/s}. \]Again, the velocity is negative because the train is moving south.

Calculate Train D's velocity

Train D travels north and covers \(24 \text{ m}\) in \(2 \text{ s}\). \[ v_D = \frac{24 \text{ m}}{2 \text{ s}} = 12 \text{ m/s}. \]The velocity is positive since the train is moving north.

Rank the velocities

Now, rank the trains by their velocity: most negative to most positive. The velocities are: - Train C: -20 m/s - Train B: -15 m/s - Train A: 10 m/s - Train D: 12 m/s The rank from most negative to most positive is: Train C, Train B, Train A, Train D.

Key concepts

Direction of Motion

One of the key aspects of understanding velocity is the direction of motion. When a moving object is considered, its velocity comprises both speed and direction. In this specific case with the trains, direction is crucial to determine the correct sign of the velocity. Here, we define north as the positive direction, and conversely, any movement towards the south is considered negative.

• Train A and Train D move north, hence they have positive velocities.
• Train B and Train C travel south, so they have negative velocities.

Understanding the direction of motion helps in assigning the correct sign to the velocities, which is essential when performing further calculations, like ranking velocities or comparing their magnitudes.

Velocity Calculation

Calculating velocity is a straightforward process that involves dividing the distance traveled by the time taken. Velocity not only tells us how fast an object is moving but also the direction of its movement. Let's break down the calculation from our exercise:

• **Train A**: Travels at a given speed of 10 m/s north.
• **Train B**: Covers 900 m in 1 minute. Convert the time to seconds to get: \[ v_B = \frac{-900 \text{ m}}{60 \text{ s}} = -15 \text{ m/s}. \ \] The negative sign reflects its southward movement.
• **Train C**: Moves south at double the speed of Train A, so: \[ v_C = -2 \times 10 \text{ m/s} = -20 \text{ m/s}. \ \] Again, the negative indicates southward direction.
• **Train D**: Travels 24 m in 2 seconds, so: \[ v_D = \frac{24 \text{ m}}{2 \text{ s}} = 12 \text{ m/s}. \ \] Positive velocity signifies it moves north.

These calculations help in understanding how fast and in which direction each train travels, forming the basis for further tasks like ranking.

Ranking Velocities

Ranking velocities means listing them in a particular order based on their magnitude and sign. In this scenario, you must arrange the velocities from the most negative (fastest southward motion) to the most positive (fastest northward motion). Here’s the ranking process, step by step:

• Train C: \[ -20 \text{ m/s} \ \] (most negative, furthest southward)
• Train B: \[ -15 \text{ m/s} \ \] (less negative, but still southward)
• Train A: \[ 10 \text{ m/s} \ \] (positive, northward)
• Train D: \[ 12 \text{ m/s} \ \] (most positive, fastest northward)

In this ordering, we have successfully arranged the trains from the slowest or most negative velocity to the fastest or most positive velocity according to their respective directions on the track.

Positive and Negative Velocities

Positive and negative velocities relate to direction relative to a chosen baseline. In mechanics, this convention helps in distinguishing opposite directions. In this exercise:

• A **positive velocity** indicates movement in the defined positive direction; north is positive for Train A and Train D, translating to velocities of 10 m/s and 12 m/s, respectively.
• A **negative velocity** shows motion in the opposite or southward direction. Train B and Train C embody this, with velocities of -15 m/s and -20 m/s, aligning with their southward paths.

This distinction is vital for physics problems and helps clarify how an object or person is moving concerning a reference direction or point. Ensuring an accurate understanding of how direction relates to positive and negative velocities can prevent errors in further dynamics calculations.