How To Find Velocity From Displacement Time Graph

Let's dive into the fascinating world of motion and graphs, specifically focusing on how to extract velocity information from a displacement-time graph. Understanding this skill is crucial for anyone studying physics, engineering, or any field that involves analyzing movement. We'll break down the concepts, provide step-by-step instructions, and explore some real-world examples to solidify your knowledge.

Understanding Displacement-Time Graphs

A displacement-time graph plots the position of an object relative to a reference point over a period. The y-axis represents displacement (often measured in meters or feet), while the x-axis represents time (usually in seconds). Analyzing this graph allows us to determine key aspects of an object's motion, including its velocity.

• Displacement: The change in position of an object. It's a vector quantity, meaning it has both magnitude and direction.
• Time: The duration over which the displacement occurs.
• Slope: The steepness of the line at any given point on the graph. Crucially, the slope of a displacement-time graph represents the velocity of the object.

The Relationship Between Slope and Velocity

The cornerstone of finding velocity from a displacement-time graph is understanding that velocity is the slope of the graph. This is derived from the fundamental definition of average velocity:

Average Velocity = (Change in Displacement) / (Change in Time)

Mathematically, this is represented as:

v = Δx / Δt

Where:

• v is the average velocity
• Δx is the change in displacement (x₂ - x₁)
• Δt is the change in time (t₂ - t₁)

On a displacement-time graph, Δx is the vertical change (rise) and Δt is the horizontal change (run). Therefore, calculating the slope (rise over run) directly gives you the velocity.

Step-by-Step Guide to Finding Velocity

Here's a detailed guide on how to determine velocity from different types of displacement-time graphs:

1. Constant Velocity (Straight Line):

This is the simplest case. A straight line on a displacement-time graph indicates constant velocity.

• Step 1: Identify Two Points: Choose any two distinct points on the straight line. Let's call them (t₁, x₁) and (t₂, x₂). It's often easiest to choose points that fall neatly on gridlines for easier reading.
• Step 2: Calculate the Change in Displacement (Δx): Subtract the initial displacement (x₁) from the final displacement (x₂): Δx = x₂ - x₁.
• Step 3: Calculate the Change in Time (Δt): Subtract the initial time (t₁) from the final time (t₂): Δt = t₂ - t₁.
• Step 4: Calculate the Slope (Velocity): Divide the change in displacement by the change in time: v = Δx / Δt.
• Step 5: Include Units: Make sure to include the correct units for velocity (e.g., meters per second (m/s), feet per second (ft/s), miles per hour (mph)).

Example:

Imagine a car moving at a constant speed. On a displacement-time graph, we find the following points:

• Point 1: (t₁ = 2 seconds, x₁ = 10 meters)
• Point 2: (t₂ = 6 seconds, x₂ = 30 meters)

1. Δx = 30 meters - 10 meters = 20 meters
2. Δt = 6 seconds - 2 seconds = 4 seconds
3. v = 20 meters / 4 seconds = 5 m/s

Therefore, the car is moving at a constant velocity of 5 m/s. A positive velocity indicates movement in the positive direction (away from the starting point). A negative velocity would indicate movement in the negative direction (towards the starting point).

2. Zero Velocity (Horizontal Line):

A horizontal line on a displacement-time graph signifies that the object is stationary.

• Interpretation: The displacement remains constant over time. Since there's no change in displacement (Δx = 0), the velocity is zero.
• Example: If a graph shows a horizontal line at x = 5 meters between t = 0 seconds and t = 10 seconds, the object is at rest at a position of 5 meters from the reference point.

3. Changing Velocity (Curved Line):

When the displacement-time graph is a curved line, it indicates that the velocity is changing. This means the object is either accelerating or decelerating. To find the instantaneous velocity at a specific point in time, you need to determine the slope of the tangent line at that point.

• Step 1: Identify the Point of Interest: Choose the specific time (t) at which you want to find the velocity.
• Step 2: Draw a Tangent Line: Carefully draw a straight line that touches the curve only at the point of interest (t). This line is called the tangent line. It represents the instantaneous slope of the curve at that point. The accuracy of your tangent line significantly impacts the accuracy of your velocity calculation.
• Step 3: Choose Two Points on the Tangent Line: Select two distinct points on the tangent line (not on the original curve). Again, choose points that are easy to read off the graph. Let's call them (t₁, x₁) and (t₂, x₂).
• Step 4: Calculate the Slope of the Tangent Line: Use the same slope formula as before: v = Δx / Δt = (x₂ - x₁) / (t₂ - t₁).
• Step 5: Include Units: Remember to include the appropriate units for velocity.

Example:

Consider a ball rolling down a ramp. Its displacement-time graph is curved. We want to find its velocity at t = 3 seconds.

1. We draw a tangent line to the curve at t = 3 seconds.
2. We choose two points on the tangent line: (t₁ = 2 seconds, x₁ = 4 meters) and (t₂ = 4 seconds, x₂ = 12 meters).
3. Δx = 12 meters - 4 meters = 8 meters
4. Δt = 4 seconds - 2 seconds = 2 seconds
5. v = 8 meters / 2 seconds = 4 m/s

Therefore, the instantaneous velocity of the ball at t = 3 seconds is approximately 4 m/s.

4. Average Velocity Over an Interval:

Sometimes, you might need to find the average velocity over a specific time interval rather than the instantaneous velocity at a single point.

• Step 1: Identify the Start and End Times: Determine the beginning (t₁) and ending (t₂) of the time interval you're interested in.
• Step 2: Find the Corresponding Displacements: Read the displacement values (x₁) and (x₂) from the graph that correspond to t₁ and t₂, respectively.
• Step 3: Calculate the Change in Displacement (Δx): Δx = x₂ - x₁.
• Step 4: Calculate the Change in Time (Δt): Δt = t₂ - t₁.
• Step 5: Calculate the Average Velocity: v_avg = Δx / Δt.

Example:

Suppose we have a runner's displacement-time graph. We want to find the average velocity between t = 10 seconds and t = 30 seconds.

1. At t₁ = 10 seconds, x₁ = 50 meters.
2. At t₂ = 30 seconds, x₂ = 200 meters.
3. Δx = 200 meters - 50 meters = 150 meters
4. Δt = 30 seconds - 10 seconds = 20 seconds
5. v_avg = 150 meters / 20 seconds = 7.5 m/s

The runner's average velocity during that 20-second interval is 7.5 m/s.

Common Mistakes to Avoid

• Confusing Displacement and Distance: Displacement is a vector (direction matters), while distance is a scalar (only magnitude matters). A displacement-time graph only shows displacement.
• Misinterpreting the Slope: Always remember that the slope represents velocity, not just speed. The sign of the slope indicates the direction of motion.
• Inaccurate Tangent Lines: Drawing accurate tangent lines is crucial for determining instantaneous velocity. Practice drawing tangent lines carefully. Use a ruler and try to make the line touch the curve at only one point.
• Incorrect Units: Always include the correct units for velocity (e.g., m/s, ft/s, mph).
• Using Points from the Curve for Tangent Line Calculations: When calculating the slope of a tangent line, make sure to select points on the tangent line itself, not on the original curve.
• Ignoring the Sign of the Velocity: The sign (+ or -) of the velocity indicates the direction of motion. Positive velocity typically means movement in the positive direction (e.g., away from the starting point), while negative velocity means movement in the negative direction (e.g., towards the starting point).

Real-World Applications

Understanding how to extract velocity from displacement-time graphs has numerous practical applications:

• Sports Analysis: Analyzing the motion of athletes during a race, jump, or throw. Coaches can use this data to improve performance.
• Vehicle Testing: Evaluating the acceleration and deceleration of cars, trains, and airplanes. Engineers use this information to design safer and more efficient vehicles.
• Robotics: Controlling the movement of robots in manufacturing, exploration, and other tasks.
• Traffic Management: Monitoring the flow of traffic on highways and in cities.
• Medical Imaging: Tracking the movement of organs or cells within the body.
• Animation and Game Development: Creating realistic motion for characters and objects in animated films and video games.

Example Problems

Let's work through some example problems to solidify your understanding.

Problem 1:

A cyclist's motion is represented by the following displacement-time data:

• (a) Calculate the cyclist's velocity between t = 0 s and t = 4 s.
• (b) Calculate the cyclist's velocity between t = 6 s and t = 8 s.
• (c) Calculate the cyclist's velocity between t = 8 s and t = 14 s.

Solution:

(a) Between t = 0 s and t = 4 s:

• t₁ = 0 s, x₁ = 0 m
• t₂ = 4 s, x₂ = 8 m
• v = (8 m - 0 m) / (4 s - 0 s) = 8 m / 4 s = 2 m/s

The cyclist's velocity is 2 m/s.

(b) Between t = 6 s and t = 8 s:

• t₁ = 6 s, x₁ = 12 m
• t₂ = 8 s, x₂ = 12 m
• v = (12 m - 12 m) / (8 s - 6 s) = 0 m / 2 s = 0 m/s

The cyclist's velocity is 0 m/s (the cyclist is stationary).

(c) Between t = 8 s and t = 14 s:

• t₁ = 8 s, x₁ = 12 m
• t₂ = 14 s, x₂ = 0 m
• v = (0 m - 12 m) / (14 s - 8 s) = -12 m / 6 s = -2 m/s

The cyclist's velocity is -2 m/s (the cyclist is moving in the opposite direction).

Problem 2:

A car's displacement-time graph is a curve. At t = 5 seconds, a tangent line is drawn. Two points on the tangent line are (4 s, 10 m) and (6 s, 30 m). What is the car's instantaneous velocity at t = 5 seconds?

Solution:

• t₁ = 4 s, x₁ = 10 m
• t₂ = 6 s, x₂ = 30 m
• v = (30 m - 10 m) / (6 s - 4 s) = 20 m / 2 s = 10 m/s

The car's instantaneous velocity at t = 5 seconds is 10 m/s.

Conclusion

Extracting velocity information from displacement-time graphs is a fundamental skill in physics and related fields. By understanding the relationship between slope and velocity, and by following the steps outlined above, you can confidently analyze these graphs and gain valuable insights into the motion of objects. Remember to pay attention to the sign of the velocity, use accurate tangent lines for changing velocities, and always include the correct units. With practice, you'll become proficient at interpreting displacement-time graphs and applying this knowledge to real-world problems.