A 1.3-kg block is pushed up against a stationary spring,...

A $1.3-\mathrm{kg}$ block is pushed up against a stationary spring, compressing it a distance of $4.2 \mathrm{~cm}$. When the block is released, the spring pushes it away across a frictionless, horizontal surface. What is the speed of the block, given that the spring constant of the spring is $1400 \mathrm{~N} / \mathrm{m}$ ?

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Key Concepts

Conservation of Energy

This principle states that the total energy in a closed system remains constant, though it may change forms. In the context of the problem, energy is transformed from the elastic potential energy stored in the compressed spring to the kinetic energy of the moving block, with no losses due to friction or external forces.

Elastic Potential Energy

Elastic potential energy is the energy stored in a deformed object, such as a compressed spring. It is quantified by the expression ½kx², where k is the spring constant and x is the displacement from equilibrium. This stored energy is released as kinetic energy when the spring returns to its original shape.

Kinetic Energy

Kinetic energy is the energy associated with an object’s motion and is given by the formula ½mv², where m is mass and v is velocity. In the problem, the block's speed is determined by the conversion of elastic potential energy into kinetic energy as it moves on a frictionless surface.

Hooke's Law

Hooke's Law describes the behavior of springs, stating that the force exerted by the spring is proportional to its displacement (F = kx). This relationship allows us to understand how much energy is stored in the spring when it is compressed, which is key for energy conversion calculations.

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