Physics: Calculate x(t), T & Kinetic Energy, Find Force & Direction of Block

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To solve the problem, the displacement function x(t) can be derived using the principles of harmonic motion, resulting in a sinusoidal equation based on the initial conditions. The period T of the oscillations can be calculated using the formula T = 2π√(m/k), leading to a value of approximately 1.25 seconds for the given spring constant. The kinetic energy of the block 4.0 seconds post-collision can be determined using the formula KE = 0.5mv², factoring in the block's velocity at that time. The force exerted by the spring on the block at t=1.2 seconds can be found using Hooke's Law, F = -kx, which also indicates the direction of the block's movement. The discussion emphasizes the application of physics concepts to analyze motion and energy in a spring-block system.
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A block of mass 0.5kg moving on a horizontal frictionless surface at 2.0 m/s collides with and sticks to a massless pan attached to the end of a horizontal ideal spring whose spring constan is 32 N/m.
a) Determine the function for x(t), the displacement from equilibrium position as a function of time.
b) What is the period T, of the subsequent oscillations?
c) What is the kinetic energy of the mass 4.0 sec after it collides with the spring?
d) What force is exerted by the spring on the block at t=1.2 sec? Which way is the block moving? Explain
 
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