Analyzing Point Motion: Coordinates & Trajectory

Hey there, physics enthusiasts! Today, we're diving into the fascinating world of kinematics, specifically focusing on how to describe the motion of a point in a plane. We'll be looking at the coordinates of a moving point, its trajectory, and how to find its position at any given time. Let's break down the problem step by step, making sure everything is crystal clear. We'll use the formulas provided to determine the equation of the trajectory, the initial coordinates, and the position of the point after a specific time. Are you ready to explore the motion of a point? Then let's get started!

Understanding the Problem: Coordinates and Motion

Our problem starts with a material point moving in the XOY plane. The coordinates of this point are changing according to specific formulas: x = -4t and y = 6 + 2t. Here, 't' represents time. These formulas tell us how the position of the point changes with time. Our primary goal is to find the equation of the trajectory, y = y(x). This equation will describe the path the point takes as it moves in the plane. We also need to find the initial coordinates (where the point starts) and the coordinates after 1 second. To solve this, we'll use our knowledge of algebra and how to manipulate equations to get the desired information. This is a classic problem in introductory physics, helping us understand the relationship between position, time, and the path of a moving object. Keep in mind that the key to solving this problem lies in understanding how the x and y coordinates are related and how they change with time. This type of problem is fundamental to understanding more complex motion scenarios later on in your studies. The connection between the parameters allows us to fully understand and define the movements of any objects.

Let's start by looking at the equations given and what they mean. The equation x = -4t tells us that the x-coordinate of the point decreases linearly with time. The equation y = 6 + 2t tells us that the y-coordinate increases linearly with time. We can see how both x and y change with time. Also, we can easily find the trajectory by eliminating the time variable, which will give us a direct relationship between x and y. This relationship will describe the actual path that the point follows in the plane. In essence, we are eliminating 't' and expressing 'y' as a function of 'x'. This will give us the equation of the trajectory. The equations we have are parametric equations where both x and y are described in terms of a parameter, in this case, the time, t. To find the equation of the trajectory, we want to eliminate this parameter and describe the relationship between x and y directly. This method can be used for more complex problems involving motion as well, where we have to understand the characteristics of the motion. Are you excited to uncover the trajectory equation? Because with it, we'll be able to draw the path of the point on the plane. Isn't that great, guys?

Finding the Equation of the Trajectory y = y(x)

Alright, let's get down to business and find that trajectory equation! Our goal is to express 'y' in terms of 'x', eliminating the time variable 't'. We have two equations: x = -4t and y = 6 + 2t. From the first equation, we can solve for 't'. Dividing both sides by -4, we get t = -x/4. Now that we have 't' in terms of 'x', we can substitute this expression into the second equation. So, substitute t = -x/4 into y = 6 + 2t. This gives us y = 6 + 2(-x/4), which simplifies to y = 6 - x/2. And there you have it! The equation of the trajectory is y = 6 - x/2. This is the equation of a straight line. This line represents the path the point takes as it moves in the XOY plane.

So, the equation y = 6 - x/2 tells us everything about the path of the point. For every value of x, we can calculate the corresponding y value, which is the position on the plane. Since it's a linear equation, we know the point will move along a straight line. The slope of the line is -1/2, and the y-intercept is 6. The slope tells us the rate at which y changes with respect to x, and the y-intercept tells us where the line crosses the y-axis. These properties of the equation provide important details on the behavior of the point.

To fully understand the motion, it is crucial to not only find the equation but also to interpret it. We can now visualize how the point is moving. As 'x' increases, 'y' decreases, indicating that the point is moving downwards and to the right along the straight line. This provides a complete picture of the motion, showing not just where the point is but also how it gets there. Isn't physics so cool?

Determining the Initial Coordinates

Next up, let's determine the initial coordinates of the moving point. Initial coordinates mean the position of the point at the very beginning of its motion, which is when time, t = 0. We can find these coordinates using the initial formulas. We know the equations x = -4t and y = 6 + 2t.

To find the initial x-coordinate, we substitute t = 0 into the equation x = -4t. This gives us x = -4(0) = 0. So, the initial x-coordinate is 0. Now, to find the initial y-coordinate, we substitute t = 0 into the equation y = 6 + 2t. This gives us y = 6 + 2(0) = 6. Therefore, the initial y-coordinate is 6. So, the initial coordinates of the moving point are (0, 6). This point represents the starting position of our point on the XOY plane. It is the precise location from which the point begins its journey.

These coordinates provide a clear reference point for the movement of the object. Knowing where the motion begins is crucial to understanding the entire path. For example, if we were to draw the line y = 6 - x/2, we can now easily see where the point starts its movement. This initial point is the foundation of our motion analysis. Remember that initial conditions are really important in physics, as they set the stage for the motion that will follow. Isn't it cool that just knowing the initial conditions, we can calculate the rest of the movement?

Finding Coordinates After 1 Second

Finally, let's determine the coordinates of the moving point after 1 second of motion. This means we're looking for the point's position when t = 1 second. Again, we'll use the given formulas to find the coordinates. We have x = -4t and y = 6 + 2t. To find the x-coordinate after 1 second, we substitute t = 1 into x = -4t. This gives us x = -4(1) = -4. So, the x-coordinate after 1 second is -4. To find the y-coordinate after 1 second, we substitute t = 1 into y = 6 + 2t. This gives us y = 6 + 2(1) = 8. Therefore, the y-coordinate after 1 second is 8. The coordinates of the moving point after 1 second are (-4, 8).

These new coordinates represent the position of the point exactly 1 second after it began moving. They allow us to track the progress of the point along its trajectory. When we see the point at (-4, 8) at t=1 second, we can say that the point has moved from its initial position (0, 6) to the new position (-4, 8). This shows how both the x and y coordinates change over a brief period. By plotting these points and the trajectory, we can visualize the point's movement more clearly. This also allows us to calculate the velocity of the point, the distance it covered, and other motion characteristics. Analyzing such details will help us understand the point's motion fully. This method can be easily extended to more complex scenarios. Physics is fun, right?

Summary and Conclusion

Alright, let's wrap things up with a quick summary of what we've accomplished today. We started with a moving point in the XOY plane, with its coordinates defined by the equations x = -4t and y = 6 + 2t. First, we found the equation of the trajectory: y = 6 - x/2. This is the equation of a straight line, meaning the point moves along a straight path. Next, we determined the initial coordinates, which were (0, 6). This is the starting point of the point's movement. After 1 second, the coordinates of the point were (-4, 8). This tells us the point's position after 1 second of motion. We've successfully analyzed the motion of the point by finding its path, starting location, and position at a specific time.

This problem gives us a good foundation for understanding more complex motion problems. It helps us visualize the concepts of kinematics and see how the x and y coordinates evolve over time. Keep in mind that in physics, we're often trying to find relationships between parameters, and in this case, the relationship between x, y, and time. These types of problems are fundamental to the concepts of physics. Keep practicing these types of problems to understand the concepts better, and you'll be well on your way to becoming a physics pro. Thanks for joining me today, guys, and keep exploring the fascinating world of physics! Do you have any questions? Don't hesitate to ask!