A 150 g particle at x = 0 is moving at 2.00 m/s in the + x - direction. As it moves, it experiences a force given by Fₓ = (0.250 N) sin (x/2.00 m) . What is the particle's speed when it reaches x = 3.14 m ?
Verified step by step guidance
1
Identify the given values: mass of the particle (m = 150 g = 0.150 kg), initial velocity (v₀ = 2.00 m/s), initial position (x₀ = 0 m), and the force function Fₓ = 0.250 N imes ext{sin}(x/2.00 m).
Understand that the force varies with position x, and use the work-energy theorem which states that the work done by the force on the particle changes its kinetic energy. The work done (W) by the force is equal to the change in kinetic energy (ΔK).
Calculate the work done by the force as the particle moves from x = 0 m to x = 3.14 m. Use the integral of the force over this distance: W = \int_{0}^{3.14} Fₓ \, dx = \int_{0}^{3.14} 0.250 \, ext{sin}(x/2.00) \, dx.
Determine the change in kinetic energy using the work-energy theorem: ΔK = K_f - K_i, where K_i is the initial kinetic energy (K_i = 0.5 \times m \times v₀^2) and K_f is the final kinetic energy (K_f = 0.5 \times m \times v_f^2).
Solve for the final speed v_f using the equation derived from the work-energy theorem: v_f = \sqrt{v₀^2 + (2 \times ΔK/m)}. Substitute the values and solve for v_f.
Recommended similar problem, with video answer:
Verified Solution
This video solution was recommended by our tutors as helpful for the problem above
Video duration:
13m
Play a video:
Was this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Newton's Second Law of Motion
Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This relationship is expressed by the equation F = ma, where F is the net force, m is the mass, and a is the acceleration. Understanding this law is crucial for analyzing how the force affects the particle's motion and speed as it travels along the x-axis.
The Work-Energy Principle states that the work done on an object is equal to the change in its kinetic energy. In this scenario, the force acting on the particle does work as it moves from x = 0 to x = 3.14 m, which will change its speed. By calculating the work done by the force over this distance, one can determine the final kinetic energy and thus the final speed of the particle.
Kinetic energy is the energy possessed by an object due to its motion, calculated using the formula KE = 0.5mv², where m is the mass and v is the velocity. As the particle moves and experiences a varying force, its kinetic energy will change. Understanding how to relate the initial and final kinetic energies will help in finding the particle's speed at the specified position.