Answer :
Final answer:
The maximum stretch of a horizontal spring-mass system can be calculated using the conservation of mechanical energy principle. Initial total mechanical energy of the system, coming from initial potential and kinetic energy, will be equal to the energy at maximum stretch which is completely potential in nature. Solving for maximum stretch will give you the answer.
Explanation:
The scenario you described relates to a physical system called a simple harmonic oscillator, specifically a spring-mass system. To find the maximum displacement or stretch of the spring, we can use the conservation of mechanical energy. The energy is conserved because the system is frictionless.
The total energy in the system is initial potential energy (PE), which is stored in the spring when compressed, and kinetic energy (KE), which the block has due to its velocity. Initial potential energy is given by the formula PE_initial = 0.5 * k * x_initial^2 where k is spring stiffness, and x_initial is initial compression. For initial kinetic energy, the formula is KE_initial = 0.5 * m * v_initial^2 where m is mass, and v_initial is initial velocity.
At maximum stretch, the spring brings the block to second halt momentarily and hence the kinetic energy of the block is zero at that moment. Hence, in this case, the total energy is completely potential in nature. Denoting the maximum stretch as x_max, we have 0.5 * k * x_max^2 = PE_initial + KE_initial. You can solve this equation to find x_max.
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