Work And Velocity Calculation: Particle Motion Problem

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Hey guys! Let's dive into a classic physics problem involving work and velocity calculations. This is a fundamental concept in mechanics, and understanding it is crucial for mastering more advanced topics. We'll break down the problem step-by-step, so you can grasp the underlying principles and apply them to similar situations. Our mission is to determine the work done by a force and the final velocity of a particle, and we will do so with a particle that's moving with a known acceleration. It sounds fun, right? Buckle up, and let's get started!

Understanding the Problem Statement

The problem presents a scenario where a particle with a mass (m) of 2.00 kg is moving under the influence of a force (Fa). The particle starts from rest at the origin and moves to a position x = 9.0 m. The particle experiences a constant acceleration (a) of 6.0 m/s². The goal is to calculate two things: first, the work done by the force (Fa) and, second, the final velocity of the particle at x = 9.0 m. To solve this, we need to bring in concepts of force, work, and energy, and also how these concepts connect to motion. Think about Newton's laws and the work-energy theorem – they are our best friends in problems like these!

Key Concepts and Formulas

Before we jump into the calculations, let's refresh the key concepts and formulas we'll be using:

  • Work Done by a Force: The work (W) done by a force (F) on an object moving a distance (d) in the direction of the force is given by:

    W = F d cos(θ)

    where θ is the angle between the force and the direction of displacement. In this case, we'll assume the force is acting in the direction of motion, so cos(θ) = 1.

  • Newton's Second Law: The force (F) acting on an object is equal to the mass (m) of the object times its acceleration (a):

    F = m a

  • Work-Energy Theorem: The work done on an object is equal to the change in its kinetic energy (KE):

    W = ΔKE = KE_final - KE_initial

    Where KE = (1/2) * m * v², with v being the velocity.

  • Kinematic Equations: Since the acceleration is constant, we can use the following kinematic equation to relate displacement (x), initial velocity (vâ‚€), final velocity (v), and acceleration (a):

    v² = v₀² + 2 * a * Δx

    where Δx is the change in position.

Step-by-Step Solution

Now, let’s apply these concepts to solve the problem.

1. Calculate the Force

First, we need to determine the magnitude of the force Fa acting on the particle. We can use Newton's Second Law:

F = m a

Given m = 2.00 kg and a = 6.0 m/s², we have:

F = (2.00 kg) * (6.0 m/s²) = 12.0 N

So, the force acting on the particle is 12.0 Newtons.

2. Calculate the Work Done

Next, we'll calculate the work done by the force as the particle moves from the origin to x = 9.0 m. Using the formula for work done:

W = F d

Here, F = 12.0 N and d = 9.0 m. Thus,

W = (12.0 N) * (9.0 m) = 108.0 J

Therefore, the work done by the force is 108.0 Joules.

3. Calculate the Final Velocity

To find the final velocity of the particle, we can use either the Work-Energy Theorem or the kinematic equation. Let's use both to check our work! :)

Method 1: Using the Work-Energy Theorem

The Work-Energy Theorem states that the work done is equal to the change in kinetic energy:

W = ΔKE = KE_final - KE_initial

Since the particle starts from rest, its initial kinetic energy (KE_initial) is 0. Thus,

W = KE_final = (1/2) * m * v²

We know W = 108.0 J and m = 2.00 kg. Solving for v:

108.0 J = (1/2) * (2.00 kg) * v²

  1. 0 = v²

v = √(108.0) ≈ 10.39 m/s

Method 2: Using Kinematic Equations

Alternatively, we can use the kinematic equation:

v² = v₀² + 2 * a * Δx

Since the particle starts from rest, v₀ = 0. Also, a = 6.0 m/s² and Δx = 9.0 m. Plugging in the values:

v² = 0² + 2 * (6.0 m/s²) * (9.0 m)

v² = 108.0

v = √(108.0) ≈ 10.39 m/s

Both methods give us the same final velocity, which is approximately 10.39 m/s. Awesome!

Putting It All Together

So, let's recap our findings:

  • The work done by the force (Fa) on the particle is 108.0 Joules.
  • The final velocity of the particle at x = 9.0 m is approximately 10.39 m/s.

We've successfully tackled this problem by applying fundamental principles of physics. The key was to understand the relationships between force, work, energy, and motion. By using Newton's Second Law, the work-energy theorem, and kinematic equations, we were able to break down the problem into manageable steps and arrive at the correct solutions.

Why This Matters

Understanding work and energy is crucial in many areas of physics and engineering. From designing machines to analyzing the motion of objects, these concepts provide a framework for understanding how forces cause changes in motion. The work-energy principle is one of the most fundamental concepts, as this principle links force and motion in a way that makes it easy to understand the effects of forces. By mastering these basic ideas, you'll be well-equipped to handle more complex problems in the future. Remember, physics is not just about memorizing formulas; it's about understanding the underlying principles and applying them creatively. Keep practicing, and you'll become a physics whiz in no time!

Practice Problems

To solidify your understanding, try these practice problems:

  1. A 3.0 kg block is pushed 15 meters across a horizontal surface by a force of 20 N. If the surface is frictionless, what is the work done by the force, and what is the final velocity of the block if it starts from rest?
  2. A 1000 kg car accelerates from rest to 25 m/s over a distance of 200 meters. Calculate the work done on the car and the average force exerted on the car.

Conclusion

Well, guys, we've journeyed through a fascinating physics problem today, delving into the concepts of work, energy, and velocity. We started with a clear understanding of the problem statement, reviewed essential formulas, and executed a step-by-step solution. By applying Newton's Second Law, the Work-Energy Theorem, and kinematic equations, we successfully calculated the work done by the force and the final velocity of the particle. Remember, physics is all about understanding the fundamental principles and applying them to solve real-world problems. Keep practicing, stay curious, and you'll be amazed at what you can achieve!