This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
ScanSolve AI Answer
Hey MR — ready when you are. To derive the equation for kinetic energy, KE = (1)/(2)mv^2, we use the work-energy theorem and Newton's second law of motion. Step 1: Define work done. Work done (W) by a constant force (F) over a displacement (S) is given by: W = F · S Step 2: Apply Newton's second law. According to Newton's second law, force (F) is the product of mass (m) and acceleration (a): F = ma Step 3: Substitute F into the work done equation. W = (ma)S Step 4: Use a kinematic equation. For an object accelerating from an initial velocity u to a final velocity v over a displacement S with constant acceleration a, the following kinematic equation applies: v^2 = u^2 + 2aS Rearranging this equation to solve for aS: 2aS = v^2 - u^2 aS = (v^2 - u^2)/(2) Step 5: Substitute the expression for aS into the work done equation from Step 3. W = m ((v^2 - u^2)/(2)) W = (1)/(2)m(v^2 - u^2) Step 6: Relate work done to kinetic energy. The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy. If an object starts from rest, its initial velocity u = 0. The work done to accelerate it to a velocity v is its kinetic energy (KE). Setting u = 0: W = (1)/(2)m(v^2 - 0^2) W = (1)/(2)mv^2 Since this work done is stored as kinetic energy: KE = (1)/(2)mv^2 The derived equation is: KE = (1)/(2)mv^2 What's next?