# Velocity-Time Graphs ## Introduction A velocity-time graph is a kinematic tool that plots an object's velocity on the vertical axis against time on the horizontal axis. This graphical representation allows one to determine the object's acceleration and displacement by analyzing specific features of the graph. ## Interpreting the Graph ### 1. Reading Velocity **Instantaneous Velocity**: The velocity at any specific moment can be read directly from the vertical axis corresponding to that time. **Direction of Motion**: - A **positive velocity** (graph above the time axis) indicates motion in the positive direction. - A **negative velocity** (graph below the time axis) indicates motion in the negative direction. - A velocity of **zero** (graph crossing the time axis) means the object is momentarily at rest. ### 2. The Slope of the Graph: Acceleration The slope of the line on a velocity-time graph represents the object's **acceleration**. This relationship is expressed as: $$ \text{Slope} = \frac{\Delta \text{Velocity}}{\Delta \text{Time}} = \text{Acceleration} $$ **Positive Slope**: A line sloping upwards indicates positive acceleration. The object is speeding up if its velocity is positive, or slowing down if its velocity is negative. **Negative Slope**: A line sloping downwards indicates negative acceleration. The object is slowing down if its velocity is positive (deceleration), or speeding up if its velocity is negative. **Zero Slope (Horizontal Line)**: A flat, horizontal line indicates zero acceleration. The object is moving at a **constant velocity**. ### 3. The Area Under the Graph: Displacement The area between the graph line and the time axis represents the object's **displacement**. - **Area Above the Axis**: Represents positive displacement (movement in the positive direction). - **Area Below the Axis**: Represents negative displacement (movement in the negative direction). - **Total Displacement**: The net area is calculated by subtracting areas below the axis from areas above the axis. - **Total Distance**: The sum of the absolute values of all areas (all areas treated as positive). ## Relationship with Other Motion Graphs | From Velocity-Time Graph | You can find | How | | :--- | :--- | :--- | | **Slope** | **Acceleration** | Calculate the gradient of the line | | **Area** | **Displacement** | Calculate the area between the line and the time axis | | **Y-axis Value** | **Instantaneous Velocity** | Read the value directly from the graph at a specific time | ## Possible Questions and Answers **Q:** What is the difference between displacement and distance on a velocity-time graph? **A:** Displacement is the *net area* under the graph, where areas below the axis are subtracted from areas above. Distance is the *total area*, where you add the absolute value of all areas, ignoring whether they are above or below the axis. **Q:** How can you tell if an object is speeding up or slowing down? **A:** An object is **speeding up** if its velocity is moving *away* from the time axis (e.g., a positive velocity becoming more positive, or a negative velocity becoming more negative). It is **slowing down** if its velocity is moving *towards* the time axis (approaching zero). ## Summary - A velocity-time graph plots **velocity** vs. **time**. - The **slope** of the graph is the **acceleration**. - The **area under the graph** is the **displacement**. - A **horizontal line** means **constant velocity** (zero acceleration). - A **straight, sloped line** means **constant acceleration**. - A **curved line** means **non-uniform (changing) acceleration**. ## References *Video lecture on velocity-time graphs*