In this problem, you will use your prior knowledge to derive one of the most important relationships in mechanics: the work-energy theorem. We will start with a special case: a particle of mass m moving in the x direction at constant acceleration a. During a certain interval of time, the particle accelerates from vi to vf, undergoing displacement s given by s=xf−xi.

Required:
a. Find the accerlation a of the particle
b. Find the force F acting on the particle
c. Find the net work W done on the particle by the external forces during the particle's motion.

Respuesta :

The kinematics and Newton's second law allow us to find the results for the questions of the motion of body are:

   A. The acceleration of the body is:  [tex]a = \frac{v^2- v_o^2}{2s}[/tex]a = v² - vo² / 2s

   B. The external force is: F =[tex]m \frac{v^2 - v_o^2}{2s}[/tex]  

   C. The work is:  W = ΔK

Kinematics studies the movement of bodies, finding relationships between the position, speed and acceleration of bodies.

A. They ask to find the acceleration of the body.

        v² = v₀² + 2a (x-x₀)

         a = [tex]\frac{v^2 -v_o^2 }{2s}[/tex]  

B. Ask for the net force on the particle.

For this we use Newton's second law that establishes a relationship between the force, the mass and the acceleration of the body.

        F = ma

     

We substitute

        F = [tex]m \ \frac{v^2-v_o^2}{2s}[/tex]

C. Work is defined as the scalar product of force times displacement.

         W = F. s

In this case the force and the displacement have the same direction,

          W = F s

We substitute

           [tex]W= m \frac{v^2-v_o^2}{2}\\W = \Delta K[/tex]

           

In conclusion, using kinematics and Newton's second law we can find the results for the questions of the motion of bodies are:

   A. The acceleration of the body is: [tex]a = \frac{v^2-v_o^2 }{2s}[/tex]

   B. The force is:   [tex]F= m \frac{v^2-v_o^2}{2s}[/tex]  

   C. The work is W = ΔK

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