Meeting Point Calculation For Two Particles Moving On The X-Axis
In the realm of physics, understanding the motion of objects is a fundamental concept. This article delves into a classic physics problem involving two particles moving along the x-axis with constant velocities. We will explore how to determine the meeting point of these particles, given their initial positions and velocities. This problem serves as a great example of applying kinematic equations and relative motion concepts. Let's dissect the problem step by step.
Problem Statement
Two particles are moving along the x-axis with uniform velocities. The first particle has a velocity of 8 m/s, while the second particle moves at 4 m/s. Initially, the first particle is located 21 meters to the left of the origin (-21 m), and the second particle is 7 meters to the right of the origin (+7 m). The objective is to determine the distance at which the two particles meet. This is a classic problem in kinematics that combines concepts of uniform motion and relative velocities. To solve this problem, we need to understand how the positions of the particles change over time and how their relative motion affects the time it takes for them to meet. The key is to set up equations that describe the position of each particle as a function of time and then solve for the time at which their positions are equal. This will give us the time of the meeting, and we can then plug this time back into either equation to find the meeting point.
Setting Up the Equations of Motion
To begin, we need to establish the equations of motion for both particles. These equations will describe their positions along the x-axis as a function of time. For an object moving with constant velocity, the position x at time t can be described by the equation:
x = x₀ + vt
where x₀ is the initial position and v is the constant velocity. For the first particle, which has an initial position of -21 m and a velocity of 8 m/s, the equation of motion is:
x₁ = -21 + 8t
This equation tells us the position of the first particle at any time t. Similarly, for the second particle, which starts at +7 m and moves at 4 m/s, the equation of motion is:
x₂ = 7 + 4t
This equation describes the position of the second particle over time. These two equations are the foundation for solving the problem. They allow us to track the positions of both particles and determine when they will occupy the same location, which is the meeting point. The next step involves using these equations to find the time at which the particles meet, and then using that time to calculate the position of the meeting point. By carefully setting up and solving these equations, we can accurately predict the outcome of this kinematic scenario.
Determining the Meeting Time
The particles will meet when their positions are equal, meaning x₁ = x₂. To find the time t at which this occurs, we set the two equations of motion equal to each other:
-21 + 8t = 7 + 4t
Now, we solve for t:
8t - 4t = 7 + 21
4t = 28
t = 7 seconds
This result tells us that the two particles will meet after 7 seconds. This is a crucial piece of information, as it allows us to pinpoint the exact moment in time when the particles occupy the same position. The meeting time is dependent on the initial positions and velocities of the particles. A larger initial separation or a smaller relative velocity would result in a longer meeting time, while a smaller initial separation or a larger relative velocity would lead to a shorter meeting time. Understanding how to calculate the meeting time is essential for solving a variety of physics problems involving moving objects. Now that we have the meeting time, we can proceed to calculate the meeting point by substituting this time back into either of the equations of motion.
Calculating the Meeting Point
Now that we know the particles meet at t = 7 seconds, we can substitute this value into either equation of motion to find the meeting point. Let's use the equation for the first particle:
x₁ = -21 + 8t
x₁ = -21 + 8(7)
x₁ = -21 + 56
x₁ = 35 meters
We can verify this result by substituting t = 7 seconds into the equation for the second particle:
x₂ = 7 + 4t
x₂ = 7 + 4(7)
x₂ = 7 + 28
x₂ = 35 meters
Both equations give us the same meeting point, which is 35 meters from the origin. This confirms our calculation and provides the final answer to the problem. The meeting point is a single, specific location where both particles are present at the same time. It is determined by the initial conditions and the velocities of the particles. In this case, the meeting point is located 35 meters to the right of the origin. Understanding how to calculate the meeting point is a fundamental skill in kinematics and is essential for analyzing the motion of objects in various scenarios.
Answer
The two particles meet at a distance of 35 meters from the origin.
Conclusion
This problem demonstrates a fundamental concept in physics: the meeting of two objects in motion. By applying the equations of motion and solving for the time and position of the meeting point, we can accurately predict the outcome of such scenarios. This problem-solving approach is applicable to a wide range of physics problems involving moving objects. Understanding these concepts is crucial for anyone studying physics or related fields. The ability to analyze and solve problems involving motion is a valuable skill that can be applied in various real-world situations. By mastering the principles of kinematics, we can gain a deeper understanding of the world around us and how objects move within it. From simple scenarios like this to more complex systems, the foundation of understanding motion lies in the ability to apply these fundamental principles.
