A particle of mass \(m\) has the potential energy function \(V(x)=m k|x|\) where \(k\) is a positive constant. What is the force when \(x>0\), and when \(x<0 ?\) Sketch the function \(V\) and describe the motion. If the particle starts from rest at \(x=-a\), find the time it takes to reach \(x=a\).

Understanding the relationship between force and potential energy is crucial to mastering concepts in classical mechanics. Potential energy, often denoted as V(x), is the energy by virtue of an object's position relative to other objects. For a particle with mass m having a potential energy function of the form V(x) = mk|x|, where k is a constant, we can find the associated force.

Force is defined as the negative gradient of the potential energy, which implies a function V(x) that increases with x will result in a force that acts in the negative x-direction, or vice versa. Mathematically, the force F(x) is determined by taking the derivative of the potential energy function with respect to position x and then negating it. For positive x values, the force is negative, indicating that the particle will feel a push back toward the origin. Conversely, for negative x values, the particle experiences a positive force, which will drive the particle toward the positive direction.

This dynamic leads to what is known as a restorative force - the force that always acts to pull the particle back towards a point of equilibrium. In our exercise, this point is at x = 0, where the potential energy is minimum and the force changes direction. This nature of the force has key implications for the motion of the particle, which we explore next.

Equations of motion are vital tools for solving classical mechanics problems, particularly when dealing with particles under the influence of constant forces. The basic equations of motion help us describe the kinematics of particles by relating displacement, velocity, acceleration, and time.

For constant acceleration, these are the classical equations we usually refer to:

• \( v = u + at \) - Final velocity after time t starting with an initial velocity u
• \( s = ut + \frac{1}{2}at^2 \) - Displacement after time t
• \( v^2 = u^2 + 2as \) - Relating final velocity to displacement

When solving our exercise, we observe that the particle's initial velocity is zero (\( u = 0 \)) and that it is subject to a constant acceleration (\( a = k \)), leading us to the second equation listed above for determining the displacement over a period of time. This type of clear, sequential use of the equations of motion simplifies the complex behavior of moving objects into manageable calculations.

By substituting known values and rearranging the equations, we can find the unknown variables such as the time it takes for a particle to move from one point to another under a constant force, demonstrating the practical utility of the equations of motion in predicting future states of motion from given conditions.

Solving problems in classical mechanics often involves a systematic approach that begins with understanding the forces at play and ends with employing mathematical equations to describe motion. The problem involving a particle acted upon by a force from a V-shaped potential well demonstrates this process.

First, we examine the nature of the force involved by analyzing the potential energy function. Recognizing that force is derived from this potential energy, we calculate the derivative to quantify the force. Next, we consider the motion of the particle under this force, bearing in mind that the direction of force will change with the sign of the position x, leading to oscillatory motion.

Finally, employing one of the key equations of motion, we unravel the duration of such oscillatory journeys. Throughout this process, it is essential to maintain a clear understanding of the physical concepts, such as potential wells and restorative forces, while applying mathematical tools, such as differentiation and the equations of motion, correctly. This combination of physical insight and mathematical prowess results in a robust method for solving a wide range of problems in classical mechanics.