# A spring with a constant of #6 (kg)/s^2# is lying on the ground with one end attached to a wall. An object with a mass of #9 kg# and speed of #4 m/s# collides with and compresses the spring until it stops moving. How much will the spring compress?

Compression:

Mechanical Energy Conservation: A system's mechanical energy is preserved when the forces operating on it are entirely conservative.

In this instance, the total mechanical energy of the system is maintained constant by increasing the potential energy in the spring by the same amount as the kinetic energy of the block decreases.

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Using the conservation of mechanical energy, the initial kinetic energy of the object is equal to the potential energy stored in the compressed spring. The initial kinetic energy of the object is ( \frac{1}{2} m v^2 ), where ( m ) is the mass of the object and ( v ) is its initial speed. The potential energy stored in the compressed spring is ( \frac{1}{2} k x^2 ), where ( k ) is the spring constant and ( x ) is the compression of the spring. Setting these equal and solving for ( x ), we find ( x = \sqrt{\frac{m v^2}{k}} ). Plugging in the given values, we get ( x = \sqrt{\frac{(9 , \text{kg})(4 , \text{m/s})^2}{6 , \text{kg/s}^2}} ). Solving this yields ( x \approx 4 , \text{m} ). Therefore, the spring will compress by approximately 4 meters.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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