# A spring with a constant of #3 (kg)/s^2# is lying on the ground with one end attached to a wall. An object with a mass of #3 kg# and speed of #6 m/s# collides with and compresses the spring until it stops moving. How much will the spring compress?

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To find the compression of the spring, you can use the conservation of mechanical energy. The spring potential energy gained equals the kinetic energy lost during the collision.

[ \frac{1}{2}kx^2 = \frac{1}{2}mv^2 ]

[ x = \sqrt{\frac{mv^2}{k}} ]

Substitute the given values:

[ x = \sqrt{\frac{(3 , \text{kg})(6 , \text{m/s})^2}{3 , \text{(kg)/s}^2}} ]

[ x = \sqrt{72} , \text{m} ]

[ x = 6\sqrt{2} , \text{m} ]

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To determine how much the spring will compress, we can use the conservation of mechanical energy.

The initial kinetic energy of the object is given by ( KE_{\text{initial}} = \frac{1}{2}mv^2 ), where ( m ) is the mass of the object (3 kg) and ( v ) is its speed (6 m/s).

The final energy of the system is entirely potential energy stored in the compressed spring, given by ( PE_{\text{final}} = \frac{1}{2}kx^2 ), where ( k ) is the spring constant (3 kg/s²) and ( x ) is the compression distance.

Since energy is conserved, ( KE_{\text{initial}} = PE_{\text{final}} ). Therefore,

[ \frac{1}{2}mv^2 = \frac{1}{2}kx^2 ]

Substitute the values:

[ \frac{1}{2} \times 3 \times (6)^2 = \frac{1}{2} \times 3 \times x^2 ]

[ 54 = 1.5x^2 ]

[ x^2 = \frac{54}{1.5} ]

[ x^2 = 36 ]

[ x = \sqrt{36} ]

[ x = 6 ]

So, the spring will compress 6 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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