Motion Equation: Find Initial Position, Velocity & Acceleration

by TextBrain Team 64 views

Hey everyone! Let's dive into a fascinating physics problem today. We've got a body moving along the OX axis, and its motion is described by a specific equation. Our mission? To figure out a few key things about its movement. We're going to find the initial position of the body, its initial velocity, and its acceleration. Plus, we'll pinpoint its position and velocity 2 seconds after we start observing it. The equation we're working with is x = -350 - 20t + 4t^2. Sounds like a fun challenge, right? Let's break it down step by step so we can understand what’s going on and how to solve it. This is crucial for understanding kinematics and how objects move under constant acceleration. So, grab your thinking caps, and let’s get started!

Understanding the Equation of Motion

First off, let's really get what this equation is telling us. The equation x = -350 - 20t + 4t^2 is a classic example of a kinematic equation, specifically describing motion with constant acceleration. You can think of 'x' as the position of the object at any given time 't'. The other terms in the equation each tell us something important about the motion. Let's break them down:

  • -350: This is the initial position of the body. It tells us where the body was located at time t = 0. In this case, it's at -350 units along the x-axis. Think of it as the starting point on our number line.
  • -20t: This term is related to the initial velocity of the body. The coefficient -20 is the initial velocity in the negative direction. So, at the beginning, the body is moving backwards at a speed of 20 units per second.
  • 4t^2: This part of the equation reveals the acceleration. The coefficient 4 is half of the acceleration. To find the actual acceleration, we need to double it, which gives us 8 units per second squared. This means the body's velocity is changing at a constant rate of 8 units per second every second.

Understanding these components is super important because it allows us to visualize the motion. We have a body starting at -350, moving backwards initially, but constantly speeding up in the opposite direction thanks to the acceleration. It's like a car initially reversing but then accelerating forward. By breaking down the equation, we've already figured out the first few parts of our problem. Now, let's move on to applying this knowledge to find specific values.

Finding Initial Position, Velocity, and Acceleration

Alright, now that we've decoded the equation, let's nail down the initial position, initial velocity, and acceleration. This is where we put our understanding into action. Remember, the equation x = -350 - 20t + 4t^2 is our roadmap. Let’s break it down piece by piece, just like cracking a code.

  • Initial Position: As we discussed earlier, the initial position is the constant term in the equation. In this case, it's -350 units. This means at time t = 0, the body is located at the -350 mark on the x-axis. Easy peasy!
  • Initial Velocity: The initial velocity is the coefficient of the t term. Here, it's -20 units per second. The negative sign tells us the body is initially moving in the negative direction along the x-axis. Think of it as moving to the left on a number line.
  • Acceleration: This is where it gets a tad tricky, but we've got this! The acceleration is related to the coefficient of the t^2 term. Remember, the coefficient (which is 4 in our case) is half of the acceleration. So, to find the actual acceleration, we multiply 4 by 2, giving us 8 units per second squared. This positive value means the body is accelerating in the positive direction, which will eventually counteract the initial negative velocity.

So, just like that, we've identified all the key parameters of the motion. The initial position is -350 units, the initial velocity is -20 units per second, and the acceleration is 8 units per second squared. These values paint a picture of the body's motion at the very start. Now, let's see how these factors influence its position and velocity after 2 seconds.

Determining Position and Velocity at t = 2 seconds

Okay, we've figured out the starting conditions. Now, let's fast forward to t = 2 seconds. Our goal is to find the body's position and velocity at this specific time. This is where we really see the power of our equation of motion. We'll use it to predict where the body will be and how fast it will be moving.

  • Position at t = 2 seconds: To find the position, we simply plug t = 2 into our equation: x = -350 - 20t + 4t^2. So, let's do it:

    • x = -350 - 20(2) + 4(2)^2
    • x = -350 - 40 + 16
    • x = -374

    So, at t = 2 seconds, the body is at -374 units along the x-axis. Notice that it has moved further in the negative direction from its initial position of -350. But don't forget, it's also accelerating in the positive direction.

  • Velocity at t = 2 seconds: To find the velocity, we need another equation. Since we have constant acceleration, we can use the equation v = vâ‚€ + at, where:

    • v is the final velocity
    • vâ‚€ is the initial velocity
    • a is the acceleration
    • t is the time

    We already know all these values! Let's plug them in:

    • v = -20 + 8(2)
    • v = -20 + 16
    • v = -4

    Therefore, at t = 2 seconds, the velocity of the body is -4 units per second. The negative sign indicates it's still moving in the negative direction, but it's slowing down because of the positive acceleration.

By calculating the position and velocity at t = 2 seconds, we've gained a snapshot of the body's motion at a specific moment. We can see how the initial conditions and the constant acceleration have influenced its movement. This is the essence of understanding kinematics – predicting motion based on given parameters. Now, let’s wrap it all up and summarize our findings.

Conclusion: Putting It All Together

Wow, we've really dug into this problem and uncovered some cool insights about the motion of this body. Let's take a moment to recap everything we've found. We started with the equation of motion, x = -350 - 20t + 4t^2, and from there, we were able to determine so much about the body's movement.

First, we identified the initial conditions: the initial position was -350 units, the initial velocity was -20 units per second, and the acceleration was 8 units per second squared. These values gave us a clear picture of where the body started and how its velocity was changing.

Then, we zoomed in on a specific moment in time, t = 2 seconds, and calculated the body's position and velocity. We found that at this time, the body was located at -374 units and had a velocity of -4 units per second. This showed us how the constant acceleration was affecting its motion over time.

This whole exercise highlights the power of kinematic equations. They allow us to describe and predict the motion of objects with constant acceleration. By understanding the components of these equations and how to apply them, we can solve a wide range of physics problems. So, whether you're analyzing the motion of a car, a ball being thrown, or even a planet orbiting a star, the principles we've discussed here are fundamental. Great job, guys, for working through this problem with me! You've tackled a real physics challenge and come out with a solid understanding of the concepts involved.