Analyzing Motion Diagrams and Calculating Velocity from Position-Time Data

Q1. Analyzing a Motion Diagram and Calculating Position and Velocity

Background

Topic: Kinematics – Position, Displacement, and Velocity

This question asks you to interpret a motion diagram and use it to determine the position and velocity of an object at various times. This is a foundational skill in introductory physics, especially when learning to connect graphical and mathematical representations of motion.

Key Terms and Formulas

• Position ($x$): The location of an object along a coordinate axis at a specific time.

• Displacement ($\Delta x$): The change in position, $\Delta x = x_f - x_i$.

• Velocity ($v$): The rate of change of position with respect to time. For constant velocity, $v = \frac{\Delta x}{\Delta t}$.

• Uniform Motion: Motion at a constant velocity (equal displacements in equal time intervals).

Step-by-Step Guidance

1. Examine the motion diagram. Notice the spacing between the dots and the direction of the arrows. Equal spacing between dots indicates constant velocity.

   

2. Identify the time intervals represented by each dot. If not explicitly labeled, assume each dot represents equal time steps (e.g., every 2 seconds).

3. To find the position at specific times (e.g., 2 s, 6 s, 10 s), count the number of intervals from the starting point and multiply by the distance per interval.

4. To calculate velocity during a time interval, use the formula:

   $v = \frac{\Delta x}{\Delta t}$

   where $\Delta x$ is the change in position over the chosen time interval, and $\Delta t$ is the duration of that interval.

5. Repeat the process for each interval specified in the sub-questions (e.g., 0–4 s, 4–6 s, 6–10 s, 10–12 s), always using the positions at the start and end of the interval.

6. Set up the calculations for each part, but stop before plugging in the final numbers. For example, for part (b):

   $v_{0-4} = \frac{x_{4} - x_{0}}{4\ \text{s}}$

Try solving on your own before revealing the answer!

Final Answers:

• (a) The positions at 2 s, 6 s, and 10 s are: $x_{2} = 2\ \text{m}$, $x_{6} = 6\ \text{m}$, $x_{10} = 10\ \text{m}$ (assuming 1 m per interval and 2 s per interval).

• (b) The velocity during the first 4 s is $v = 1\ \text{m/s}$.

• (c) The velocity from 4 s to 6 s is $v = 1\ \text{m/s}$.

• (d) The velocity from 6 s to 10 s is $v = 1\ \text{m/s}$.

• (e) The velocity from 10 s to 12 s is $v = 1\ \text{m/s}$.

All intervals show the same velocity, confirming the motion is uniform. The positions increase by equal amounts in equal time intervals, which is characteristic of constant velocity motion.