Question

Learning Goal: To understand the meaning andpossible applications of the work-energy theorem.

In this problem, you will use your prior knowledge to derive one ofthe most important relationships in mechanics: the work-energytheorem. We will start with aspecial case: a particle of massmoving in the x directionat constant acceleration.During a certain interval of time, theparticle accelerates from $v_initial$ to $v_final$ , undergoing displacement given by .part A.Find the acceleration of the particle.Express the acceleration in terms of $v_initial$ , $v_final$ , and .part BFind the net force acting on the particle.Express your answer in terms ofand .part cFind the net work done on the particle by the externalforces during the particle'smotion.Express your answer in terms ofand .part DSubstitute for from Part B in the expression forwork from Part C. Then substitutefor from therelation in Part A. This will yield an expression for thenet work done on the particle by the external forces during the particle'smotion in terms of mass and the initial and final velocities.Givean expression for the work in terms of thosequantities.Express your answer in terms of, $v_initial$ , and $v_final$ .part EFind the net work done on the particle by theexternal forces during the motion ofthe particle in terms of the initial and final kineticenergies.Express your answer in terms of $K_initial$ and $K_final$ .part FEvaluate the integral .Express your answer in terms of, $v_initial$ , and $v_final$ .part GA particle moving in the x directionis being acted upon by a net force $F(x)=Cx^2$ , for some constant .The particle moves from to . What is , the change in kinetic energy of the particleduring thattime?Express your answer in terms ofand.

Answer 1

a)accelerationa =( $v_final$ ² - $v_initial$ ² )/ 2Sb)net force isF =m ac) work doneW = F .Sd) workW = (1/2)m[ $v_final$ ² - $v_initial$ ²]e) workW = $K_final$ - $K_initial$ f) W = m W = m [ V²_final -V²_initial ] /2g)The change in KE is= (C /3) [9L³ - L³ ]= 8CL³ /3