How To Do Velocity Vectors In Desmos

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Post on Feb 09, 2025

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How To Do Velocity Vectors in Desmos

Desmos, the popular online graphing calculator, offers a powerful way to visualize velocity vectors, providing a dynamic understanding of motion. This guide will walk you through how to represent and manipulate velocity vectors in Desmos, enhancing your understanding of physics and calculus concepts.

Understanding Velocity Vectors

Before diving into Desmos, let's refresh the concept of velocity vectors. A velocity vector represents both the speed and direction of an object's motion. It's a vector quantity, meaning it has both magnitude (speed) and direction. Unlike speed (a scalar quantity), velocity tells us how fast and where an object is moving.

For example, a velocity vector of <3, 4> would represent an object moving 3 units horizontally and 4 units vertically per unit of time. The magnitude (speed) can be calculated using the Pythagorean theorem: √(3² + 4²) = 5 units per unit of time.

Representing Velocity Vectors in Desmos

Desmos utilizes the notation <x, y> to represent vectors. To plot a velocity vector, you'll define it as a point and then visually represent it as an arrow.

Here's how you can do it:

1. Define the vector: Let's say our velocity vector is v = <2, 3>. In Desmos, you don't need to explicitly label it 'v', but it helps for clarity.

2. Plot the starting point: Choose a starting point for your vector. For simplicity, we'll often use the origin (0,0). You would type this into Desmos as a point: (0, 0).

3. Plot the end point: Add the vector components to your starting point. For our vector <2, 3>, starting at (0,0), the endpoint will be (0+2, 0+3) = (2, 3). Type this into Desmos as another point: (2, 3).

4. Create the arrow: Desmos doesn't directly draw vectors as arrows. We achieve this by using a line segment connecting the start and end points, visually representing the vector. Desmos will connect the points (0,0) and (2,3) if they are entered as individual points.

Visualizing Time-Dependent Velocity

The real power of Desmos is its ability to handle time-dependent velocities. Imagine an object whose velocity changes over time. This is easily visualized using parameters and lists.

Let's consider the velocity vector v(t) = <t, 2t>. This means the horizontal component increases linearly with time, and the vertical component increases twice as fast.

1. Use parameters: Introduce a slider for time 't' in Desmos by typing t = 0. You can adjust the range and step size of the slider.

2. Define the end point as a function of time: The end point of the vector will now be a function of 't'. In Desmos, enter the point (t, 2t).

3. Observe the change: As you adjust the slider for 't', you'll see the endpoint of the vector move, clearly visualizing how the velocity changes over time.

Advanced Techniques: Combining Vectors & Acceleration

Desmos allows for more complex scenarios. You can:

• Add vectors: If you have multiple velocity vectors, you can add them to find the resultant velocity.
• Introduce acceleration: By integrating acceleration vectors, you can model changing velocities over time with even greater accuracy.

For instance, if you have a constant acceleration vector a = <1, -1>, you could update the velocity vector as v(t) = <t, 2t - t²> (this is a simplified example). This will visually demonstrate a projectile motion pattern.

Conclusion

Desmos offers a user-friendly and powerful environment for visualizing velocity vectors. By understanding the basic principles of vector representation and leveraging Desmos' capabilities with parameters and lists, you can create dynamic visualizations of motion, furthering your understanding of physics and related mathematical concepts. Remember to experiment with different scenarios and parameters to explore the possibilities!