A spring of constant \(k,\) compressed a distance \(x,\) is used to launch a mass \(m\) up a frictionless slope at angle \(\theta .\) Find an expression for the maximum distance along the slope that the mass moves after leaving the spring.

Kinetic energy is defined as the energy an object possesses due to its motion. For any moving object with mass m and velocity v, the kinetic energy (KE) can be calculated using the formula \( KE = \frac{1}{2}mv^2 \). When it comes to our spring-launched mass, the kinetic energy at the moment it leaves the spring is pivotal. This energy is what propels the mass upwards along the frictionless slope.

At the commencement of motion, all energy stored in the spring (elastic potential energy) is converted into kinetic energy. Hence, for the mass to leave the spring with maximum kinetic energy, it must utilize all the energy stored within the compressed spring. This is represented mathematically by \( KE_{initial} = \frac{1}{2}kx^2 \), where k is the spring constant and x is the compression distance of the spring. The conversion from the potential energy in the spring to the kinetic energy of the mass is a demonstration of the conservation of energy principle, an underlying concept in physics that dictates that energy cannot be created or destroyed, only transformed from one form to another.

Potential energy (PE), in contrast to kinetic energy, is the energy stored by an object due to its position or state. For an object at a height h above the ground, the gravitational potential energy can be given by \( PE = mgh \), where m is the mass of the object, g is the acceleration due to gravity, and h is the height above the reference point.

In the context of our problem, as the mass moves along the frictionless slope to its peak point, the kinetic energy it started with is steadily converted to gravitational potential energy. The highest point reached by the mass correlates to its maximum potential energy. This energy is described by \( PE_{final} = mgd \sin(\theta) \), with d being the maximum distance traveled along the slope and \(\theta\) the angle of the slope. Understanding the interchange between kinetic and potential energy is crucial for solving problems related to motion under the influence of gravity.

The spring constant, denoted as k, is a measure of a spring's stiffness. It is defined as the force required per unit length of extension or compression. Mathematically, it is expressed by Hooke's Law: \( F = -kx \), where F is the force applied to the spring and x is the displacement from the spring's equilibrium position. The negative sign indicates that the force exerted by the spring is in the opposite direction to the displacement.

A spring with a larger spring constant is stiffer, and thus more force is needed to compress or stretch it by a given amount. In the launch scenario, a higher spring constant implies that for the same compression distance x, the spring can store more energy, which is then transferred to the mass as kinetic energy when released. Therefore, the spring constant plays a direct role in determining the initial kinetic energy of the mass and affects the maximum distance the mass can travel along the slope.