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?

검증된 단계별 안내

Step 1: Recognize that the work-energy principle applies here. The work done by the force on the particle will result in a change in its kinetic energy. The work-energy theorem states: ΔK = W, where ΔK is the change in kinetic energy and W is the work done by the force.

Step 2: Write the expression for kinetic energy. The kinetic energy of the particle is given by K = (1/2)mv², where m is the mass of the particle and v is its velocity. The change in kinetic energy is ΔK = (1/2)m(v_f² - v_i²), where v_f is the final velocity and v_i is the initial velocity.

Step 3: Calculate the work done by the force. The work done by a variable force is given by W = ∫ Fₓ dx, where Fₓ is the force as a function of position x. Substitute Fₓ = (0.250 N) sin(x / 2.00 m) into the integral: W = ∫ (0.250 N) sin(x / 2.00 m) dx, with limits of integration from x = 0 to x = 3.14 m.

Step 4: Solve the integral for work. The integral of sin(ax) is (-1/a) cos(ax), where a is a constant. Here, a = 1 / (2.00 m). Perform the integration and evaluate the result at the limits x = 0 and x = 3.14 m.

Step 5: Use the work-energy theorem to find the final speed. Substitute the calculated work W and the given mass m = 150 g = 0.150 kg into the equation ΔK = W. Solve for v_f using v_f² = v_i² + (2W / m), where v_i = 2.00 m/s. This will give the final speed of the particle.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

13m

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 mathematically as 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 acting on the particle affects its motion and speed.

Work-Energy Principle

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. This principle allows us to relate the force to the particle's speed by calculating the work done over the distance traveled.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, calculated using the formula KE = 0.5mv², where m is the mass and v is the velocity of the object. As the particle moves and experiences a force, its kinetic energy will change, which can be analyzed to determine its final speed at a given position. Understanding kinetic energy is essential for solving the problem and finding the particle's speed at x = 3.14 m.