Smallest Displacement In Linear Motion Analyzing Equations Of Motion
In physics, understanding the motion of objects is fundamental, and one of the key concepts in describing motion is displacement. Displacement refers to the change in position of an object and is a vector quantity, meaning it has both magnitude and direction. When analyzing motion along a single axis, such as the Ox axis, we can use equations of motion to determine the displacement of different objects over the same time interval. This article delves into the analysis of several equations of motion to identify which body undergoes the smallest displacement within a given timeframe. We will explore how to interpret these equations and apply them to solve practical problems, providing a comprehensive understanding of linear motion.
The study of motion, often called kinematics, is a cornerstone of classical mechanics. Understanding how objects move, their velocities, and their accelerations allows us to predict their future positions and interactions. Equations of motion, which are mathematical expressions that describe the position of an object as a function of time, are invaluable tools in this endeavor. These equations typically take the form of linear equations when dealing with constant velocity motion, and quadratic or more complex forms when acceleration is involved. Our focus here is on linear equations, which represent uniform motion where the velocity remains constant. These equations are straightforward yet powerful in illustrating the fundamental principles of displacement and motion. By analyzing the coefficients within these equations, we can quickly ascertain the initial position and velocity of an object, and subsequently calculate its displacement over any time interval.
In this article, we will dissect several equations of motion to determine which represents the smallest displacement over an identical time period. This requires a careful examination of the coefficients in each equation, which dictate the object's velocity. The core concept we will use is that displacement is the product of velocity and time when the velocity is constant. Therefore, the equation with the smallest velocity magnitude will result in the smallest displacement magnitude over a given time interval. This analysis not only helps in solving specific problems but also builds a solid foundation for understanding more complex motion scenarios. We will start by defining the basic equation of motion and then apply this understanding to the given set of equations, systematically identifying the equation that yields the smallest displacement. Through this process, we will reinforce the relationship between velocity, time, and displacement, which is crucial for solving a wide range of physics problems.
In order to determine the body with the smallest displacement, it's crucial to understand the general form of the equation of motion for an object moving with constant velocity along the Ox axis. The general equation is given by:
x = x₀ + vt
Where:
xis the final position of the object.x₀is the initial position of the object.vis the velocity of the object (which can be positive or negative).tis the time elapsed.
Displacement (Δx) is defined as the change in position, which can be calculated as:
Δx = x - x₀ = vt
The magnitude of the displacement, which is the absolute value of Δx, tells us the distance the object has moved, regardless of direction. Therefore, the magnitude of the displacement is given by:
|Δx| = |vt|
To find the body with the smallest displacement for the same interval of time, we need to compare the magnitudes of |v| for each body, because t is constant. The smaller the magnitude of v, the smaller the magnitude of the displacement.
The equation of motion is a fundamental concept in classical mechanics, providing a clear and concise way to describe how an object's position changes over time. In the context of uniform motion, where the velocity remains constant, this equation takes a linear form, making it relatively easy to interpret and apply. The equation x = x₀ + vt encapsulates all the essential information about the object's motion: its starting point (x₀), its speed and direction (v), and the time elapsed (t). By understanding each component of this equation, we can predict the object's position at any given time and calculate its displacement over any interval.
The initial position, x₀, simply represents where the object was located at the beginning of the observation period (t=0). The velocity, v, is the rate at which the object's position changes and is a crucial factor in determining the displacement. A positive velocity indicates movement in the positive direction along the Ox axis, while a negative velocity indicates movement in the opposite direction. The magnitude of the velocity signifies the speed of the object. The time elapsed, t, is the duration over which the motion is observed. The longer the time, the greater the potential displacement, given a constant velocity. The displacement, Δx, is the net change in position, calculated as the final position minus the initial position. It's a vector quantity, meaning it has both a magnitude and a direction. In our analysis, we are particularly interested in the magnitude of the displacement, which tells us the total distance covered, irrespective of direction.
When comparing the displacements of different objects over the same time interval, the velocity becomes the key factor. Objects with smaller velocities will naturally have smaller displacements. Therefore, our task in identifying the smallest displacement boils down to comparing the magnitudes of the velocities in the given equations of motion. We will carefully examine the coefficient of t in each equation, as this coefficient represents the velocity. The equation with the smallest velocity magnitude will correspond to the object with the smallest displacement magnitude over the specified time interval. This method provides a straightforward way to solve problems involving comparative motion and highlights the importance of understanding the components of the equation of motion.
Now, let's analyze the given equations:
A) x = 20 + 3t
B) x = 40 + t
C) x = 40 - 4t
D) x = 40 - 2t
E) x = 10 + 6t
From these equations, we can identify the velocities (v) for each body:
A) v = 3
B) v = 1
C) v = -4
D) v = -2
E) v = 6
To find the smallest displacement, we need to find the smallest magnitude of velocity:
|A) |v| = |3| = 3
|B) |v| = |1| = 1
|C) |v| = |-4| = 4
|D) |v| = |-2| = 2
|E) |v| = |6| = 6
The smallest magnitude of velocity is 1, which corresponds to equation B.
Analyzing equations of motion involves extracting pertinent information and comparing it across different scenarios. In this case, we are given five equations, each representing the position of a body as a function of time. The goal is to identify which body undergoes the smallest displacement over an identical time interval. The key to this analysis lies in recognizing that each equation is in the form x = x₀ + vt, where x is the final position, x₀ is the initial position, v is the velocity, and t is the time elapsed. By dissecting each equation, we can isolate the velocity component, which is crucial for determining the displacement.
The velocity term in each equation is the coefficient of t. For instance, in equation A) x = 20 + 3t, the velocity is 3. This indicates that the body is moving in the positive direction along the Ox axis at a rate of 3 units per unit of time. Similarly, in equation C) x = 40 - 4t, the velocity is -4, meaning the body is moving in the negative direction at a rate of 4 units per unit of time. The sign of the velocity is important because it tells us the direction of motion, but when we are interested in the magnitude of displacement, we focus on the absolute value of the velocity.
Once we have identified the velocities for each body, we compare their magnitudes. The magnitude of velocity is simply the absolute value of the velocity, stripping away the sign. This gives us the speed of the body, which is the rate at which it is moving without considering direction. The body with the smallest speed will have the smallest displacement over the same time interval. In our analysis, we calculated the magnitudes of the velocities for each body and found that the smallest magnitude is 1, corresponding to equation B) x = 40 + t. This means that the body described by equation B moves the slowest and therefore experiences the smallest displacement within a given time period. This step-by-step approach highlights the importance of careful observation and comparison in solving physics problems, emphasizing how a clear understanding of fundamental concepts can lead to accurate conclusions.
Therefore, the body described by equation B) x = 40 + t has the smallest displacement for the same interval of time.
In conclusion, determining the smallest displacement among several moving bodies requires a clear understanding of the equations of motion and the factors that influence displacement. We've seen that displacement is directly proportional to velocity when the time interval is constant. Therefore, identifying the smallest displacement involves comparing the magnitudes of the velocities of the bodies in question. By extracting the velocity component from each equation and comparing their absolute values, we can effectively determine which body moves the least during the same time period. This method highlights the importance of breaking down complex problems into simpler components and applying fundamental physics principles to arrive at a solution.
The analysis we performed underscores the significance of the equation of motion x = x₀ + vt in describing linear motion. This equation provides a straightforward way to represent the position of an object as a function of time, making it easy to extract information about the object's initial position and velocity. The velocity, in particular, plays a crucial role in determining displacement. A smaller velocity magnitude implies a smaller displacement for a given time interval. This relationship is fundamental in understanding how objects move and interact, and it forms the basis for more advanced concepts in mechanics.
The process of comparing equations of motion and identifying key parameters is a valuable skill in physics problem-solving. It requires careful attention to detail and a solid understanding of the underlying principles. By practicing these types of analyses, students and professionals can develop a deeper appreciation for the elegance and power of physics in describing the world around us. The ability to break down equations, extract relevant information, and make comparisons is essential not only in physics but also in many other scientific and engineering disciplines. Thus, mastering these skills provides a strong foundation for tackling a wide range of problems and challenges.
Final Answer: The final answer is B.
