Determining A Mobile's Equation And Nature In Physics

Hey everyone! Today, we're diving into a cool physics problem: figuring out the equation and characteristics of a moving object, or what we call a 'mobile' in French. The specific situation we're looking at involves a mobile with its position defined by the following equations: x = 5t^2 and y = 4t + 3. Let's break it down step by step to understand how we can describe its movement and what kind of path it follows. Don't worry, it's easier than it sounds!

Understanding the Given Equations

First off, let's get familiar with what we're dealing with. The equations x = 5t^2 and y = 4t + 3 describe the position of our mobile in a 2D space. Here, 'x' and 'y' represent the coordinates of the mobile at any given time 't'. Think of 't' as the time elapsed since the mobile started moving. Each 't' value gives us a specific (x, y) location, like a snapshot of where the mobile is at that moment. The equation x = 5t^2 tells us how the horizontal position of the mobile changes over time. Notice that 'x' is proportional to the square of time. This means that as time goes on, the 'x' position increases at an increasing rate. The equation y = 4t + 3 tells us about the vertical position. Here, 'y' changes linearly with time. The '+3' means that the mobile starts at a vertical position of 3 units when t=0, and the '4t' indicates that the vertical position increases at a constant rate as time passes. The '4' represents the vertical velocity component. In simple terms, the mobile is moving both horizontally and vertically, and these equations tell us exactly how these movements are happening. It's like having two independent clocks ticking, one for the x-axis and one for the y-axis, influencing the final position of the object. Understanding these individual components is crucial for describing the overall motion and nature of the path. What this means is that we can understand where the mobile will be at any given moment. So, for instance, if t=1 second, we can calculate the position of the mobile, and we will know that x=5 and y=7.

Now, let's see what we can do with these equations. We're aiming to find out two key things: the equation of the path the mobile takes and the nature of this path. What shape does it follow? Is it a straight line, a curve, or something else?

Determining the Equation of the Path

To find the equation of the path, we want to eliminate the time variable, 't', from our two equations. This will give us a direct relationship between 'x' and 'y', which defines the path. Here's how we can do it.

We have: x = 5t^2 and y = 4t + 3.

First, let's solve the second equation for 't'.

y = 4t + 3 y - 3 = 4t t = (y - 3) / 4

Now, substitute this value of 't' into the first equation:

x = 5t^2 x = 5 * ((y - 3) / 4)^2

Simplify the equation:

x = 5 * (y^2 - 6y + 9) / 16 16x = 5y^2 - 30y + 45

Now, rearrange the terms to get a more standard form:

5y^2 - 30y - 16x + 45 = 0

So, the equation of the path of the mobile is 5y^2 - 30y - 16x + 45 = 0. This is a quadratic equation in terms of 'y' and 'x'. The next step is to identify what kind of path it describes. We've successfully eliminated time and established a direct link between the horizontal and vertical positions of our mobile. This single equation encapsulates the complete trajectory, revealing the specific relationship between x and y.

This is an important step, because it allows us to predict where the mobile will be at any point in its path. The equation we found is the secret ingredient to describing this mobile's movement fully. What it does, in essence, is define the precise relationship between the x and y coordinates. Because the equation of the path is expressed without 't', we can plug in any 'x' value and solve for the corresponding 'y' value to find the position of the mobile. This gives us a clear understanding of the trajectory that the mobile follows. Each (x, y) point that satisfies this equation lies on the path of the mobile.

Identifying the Nature of the Path

Now, let's figure out the nature of the path described by the equation 5y^2 - 30y - 16x + 45 = 0. To identify the type of path, we can rewrite the equation and analyze it. If you are comfortable with calculus, the derivative will tell you a lot about the behavior of the mobile. Let's try to get the equation into a more recognizable form. The equation we have includes a y^2 term and an x term, which suggests a specific type of curve. The general form is a quadratic equation. The presence of the y^2 term and the x term indicates the equation represents a parabola. The parabola can open either horizontally or vertically depending on the coefficients. In this case, because the y variable is squared, we are dealing with a horizontal parabola, meaning that the parabola opens either to the left or to the right. We can rewrite the equation to confirm this. Let's complete the square for the y terms to see it more clearly. It can be done by the following steps:

First, isolate the y terms:

5y^2 - 30y = 16x - 45

Factor out the 5 from the y terms:

5(y^2 - 6y) = 16x - 45

Complete the square inside the parenthesis. To do this, take half of the coefficient of the y term (-6), square it ((-3)^2 = 9), and add it inside the parenthesis. To keep the equation balanced, we also add 5 * 9 (since we factored out a 5) to the right side:

5(y^2 - 6y + 9) = 16x - 45 + 5 * 9 5(y - 3)^2 = 16x - 45 + 45

Simplify:

5(y - 3)^2 = 16x

Divide both sides by 16 to get the standard form:

(y - 3)^2 = (16/5)x

This is the standard form of a horizontal parabola. The vertex is at the point (0,3). The parabola opens to the right. This form clearly shows that the path is a parabola with a horizontal axis of symmetry. The vertex of the parabola is at the point where the squared term is zero, and the x value determines the direction in which it opens. This means the path our mobile takes is not a straight line, but a curved path that resembles a smile. This shape results directly from how x and y change with time as defined by the original equations. So, our mobile is tracing out a parabolic path!

Conclusion

Alright, guys, that's a wrap! We've successfully determined the equation of the path (5y^2 - 30y - 16x + 45 = 0) and identified the nature of the path as a horizontal parabola. This means our mobile, as it moves, traces a curved path, with its movements governed by those original equations, x = 5t^2 and y = 4t + 3. It's all about understanding how x and y change in relation to each other and how the time variable, 't', affects these positions. Physics can be so cool, right? Keep practicing, and you'll be able to solve these problems like a pro!

Keep in mind that this is a basic example. The concepts you learned here are fundamental in physics, and they are the basis for understanding motion and trajectories. As you study more, you'll find more complex problems that build on these fundamental concepts.

If you liked this and want to learn more, go ahead and explore the world of physics. There's a whole lot more to discover! Keep practicing and have fun!