The position-time graph shows the motion of three objects (labeled 1, 2, 3).

Question

A position-time graph.Short description, A position-time graph.,Long description,
The graph is entitled Position versus Time, with time on the x-axis and position on the y-axis. The positions over time of three objects are plotted as they take different paths between the same two points, Point A and Point B. Point A is high on the y-axis, and point B is low on the y-axis.

Object one’s motion is a curved arc, with a higher velocity at the start of the curve, and a lower velocity at the end.
Object two’s motion is a straight line, with velocity remaining constant between the points.
Object three’s motion is a curved arc, with a lower velocity at the start of the curve, and a higher velocity at the end.

Question
How are the average velocities of the objects related to each other?

Answer options with 4 options
A.
v Subscript 1 Baseline equals v Subscript 2 Baseline equals v Subscript 3 Baseline

C.
v Subscript 1 Baseline is less than v Subscript 2 Baseline which is less than v Subscript 3 Baseline

B.
v Subscript 1 Baseline is greater than v Subscript 2 Baseline which is greater than v Subscript 3 Baseline

D.
v Subscript 1 Baseline equals v Subscript 3 Baseline which is greater than v Subscript 2 Baseline

Answer 1

To determine the relationship between the average velocities of the three objects based on the description of their motion on the position-time graph, we can analyze their paths qualitatively:

1. Object 1: Its motion is represented by a curved arc where the velocity starts high and decreases. This indicates that while the object moves quickly at the beginning, it slows down by the end.

2. Object 2: This object moves in a straight line, maintaining a constant velocity throughout the motion. Thus, its average velocity can be directly measured as the slope of the line connecting Points A and B.

3. Object 3: This object also follows a curved arc but starts with a lower velocity and accelerates towards the end. So, it begins slower and ends with a higher velocity.

Considering average velocities:

• The average velocity of Object 2 (constant velocity) will be in the middle because it does not accelerate or decelerate; it maintains the same speed.
• Object 1 starts fast and slows down, likely resulting in a lower average velocity than that of Object 2.
• Object 3 starts slow but accelerates, suggesting its average velocity can be higher than that of Object 2.

Thus, we can set up the relation as follows:

• \( v_1 < v_2 \) (Object 1 has a lower average velocity due to deceleration)
• \( v_2 \) is in the middle due to constant velocity motion
• \( v_3 > v_2 \) (Object 3 has a higher average velocity due to acceleration)

Putting it all together, we conclude that the correct relationship is:

D. \( v_1 < v_2 < v_3 \)

However, since "less than" and "greater than" are not included in the answer options as such, we must match with the available choice. The option that resembles this arrangement best is:

D: \( v_1 \) equals \( v_3 \) which is greater than \( v_2 \).

This captures the fact that \( v_3 \) is accelerating and can be greater than both \( v_1 \) and \( v_2 \), but it does not note that \( v_1 < v_3 \), which is the true case.

In conclusion, while none of the options provide an accurate description of the relationships, D is the best option. Thus, my answer is:

D. \( v_1 \) equals \( v_3 \) which is greater than \( v_2 \).