Meeting Point Of Vehicles: A Physics Problem

Hey guys! Let's dive into a classic physics problem involving two vehicles in uniform motion. We'll figure out when and where they meet. This kind of problem is super useful for understanding how speed, distance, and time all relate to each other. It's a fundamental concept in physics, and once you get the hang of it, you'll be able to solve many similar problems. So, buckle up, grab your calculators, and let's get started! We're going to look at a scenario where two vehicles are cruising along a highway, each with its own constant speed. One starts at the beginning, and the other has a head start. Our mission? To find out when and where they finally cross paths. This problem is all about understanding the relationship between distance, velocity, and time, which is a core concept in physics. By working through this example, you'll build a solid foundation for tackling more complex motion problems. It's like learning the basics of a video game before you start playing the harder levels. Ready to learn how to calculate when two vehicles meet? Let's get into it.

Setting the Stage: The Vehicles and Their Journey

Alright, let's set the scene. Imagine two vehicles speeding down a straight highway. The first vehicle, which we'll call Vehicle A, starts its journey from the origin – that's the zero-kilometer mark. Vehicle A is moving at a steady pace of 90 kilometers per hour (km/h). That's a pretty good clip! Now, the second vehicle, Vehicle B, has a bit of a head start. It begins its trip from the 300-kilometer mark. And here's the twist: Vehicle B is moving in the opposite direction of Vehicle A. Its velocity is -110 km/h. The negative sign simply tells us that Vehicle B is moving towards the origin, meaning it's traveling in the opposite direction of Vehicle A. So, we have one car starting at zero and moving forward, and the other starting further down the road and coming back towards the start. What we want to determine is when the two vehicles will finally meet. This requires us to put together a couple of equations and solve for the time and distance at which the vehicles intersect. It's all about understanding the relationship between distance, velocity, and time. Let's see how we can find the meeting point of our two vehicles on the road!

To solve this, we'll need to recall the basic equation for uniform motion: distance = velocity × time (d = v × t). We can use this to set up equations for each vehicle. For Vehicle A, the distance it covers will be directly related to its speed and the time elapsed since it started. For Vehicle B, things are a little different because of its initial position and its negative velocity. The equation for Vehicle B will account for its starting point and the fact that it's moving towards the origin. Understanding the setup is crucial, because it is the framework for the mathematical solution. We'll use these equations to find the time when both vehicles are at the same position, which is when they meet. The journey of each vehicle is defined by its starting point, velocity, and the time it travels. To simplify things, we assume the motion is linear and that the speeds are constant. Using the basic equation, we can then determine the time it takes for the vehicles to collide. We will find a time where both cars have the same position coordinate. With a bit of algebra, we'll find the point where these cars collide.

The Equations of Motion: Pinpointing the Meeting Place

Now, let's translate our scenario into mathematical equations. This is where the magic happens! We'll use the equation of motion (d = v × t) to describe each vehicle's journey. For Vehicle A, since it starts at the origin (position 0), its position (d_A) at any given time (t) will be simply its velocity (90 km/h) multiplied by the time: d_A = 90t. This means that the distance traveled by Vehicle A increases linearly with time. For Vehicle B, we need to consider its initial position (300 km) and its velocity (-110 km/h). The equation for Vehicle B's position (d_B) at any time (t) is: d_B = 300 - 110t. The initial position is positive because the car starts at the 300 km mark. The negative sign for the velocity indicates it's moving toward the origin. Now, these equations are key. They describe where each vehicle is at any point in time. When the vehicles meet, their positions will be the same. So, to find the time and location of their meeting, we need to set d_A equal to d_B and solve for t. Once we find the time (t), we can plug it back into either equation to find the distance (d) where they meet. This is the core of the problem. It's about finding a common point in space and time for both vehicles. By equating their positions and solving for time, we uncover the precise moment of their encounter. So, we're using the equations to find when the distance of A equals the distance of B.

Think of it like this: The equations are like two lines on a graph. The point where the lines intersect is the point where the vehicles meet. It’s that intersection that we are trying to discover. The first equation represents Vehicle A's progress, and the second represents Vehicle B's approach. Where the equations meet is the solution to our problem. We combine the equation of motion with a little bit of algebra to find the spot where the vehicles meet.

Solving for Time and Position: The Grand Finale

Let's solve the equations! To find the time when the vehicles meet, we set the two position equations equal to each other: 90t = 300 - 110t. Now, we'll do some algebra. First, add 110t to both sides to get all the 't' terms on one side: 90t + 110t = 300. This simplifies to 200t = 300. Next, to isolate 't', divide both sides by 200: t = 300 / 200, which gives us t = 1.5 hours. So, the vehicles will meet after 1.5 hours. Now that we have the time, we can calculate the position where they meet. We can plug the time (t = 1.5 hours) into either of the position equations. Let’s use Vehicle A's equation: d_A = 90t. Plugging in t = 1.5 hours, we get d_A = 90 * 1.5 = 135 km. Therefore, the vehicles meet at the 135-kilometer mark. So, after 1.5 hours, Vehicle A will have traveled 135 kilometers, and Vehicle B, starting at 300 km, will have traveled towards the origin and will also be at the 135 km mark. This is the point where they meet. It's pretty awesome, right? We started with two vehicles, different speeds, and a head start, and we were able to pinpoint exactly when and where they would cross paths. Now we know the exact time and location where the vehicles meet. Solving the system of equations gives us these two parameters, defining when and where the vehicles meet. We have successfully completed the calculation of the meeting point for both vehicles!

This kind of problem shows how powerful the equation of motion really is. Understanding this method will help you analyze other physics problems involving moving objects and collisions.