# A spring with a constant of #12 (kg)/s^2# is lying on the ground with one end attached to a wall. An object with a mass of #6 kg# and speed of #4 m/s# collides with and compresses the spring until it stops moving. How much will the spring compress?

The spring will compress by

The spring constant is

The kinetic energy of the object is

This kinetic energy will be stored in the spring as potential energy.

So,

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The potential energy stored in the spring when compressed is equal to the kinetic energy of the object before the collision.

The kinetic energy of the object before the collision is given by: [ KE = \frac{1}{2}mv^2 ]

Where:

- ( m ) is the mass of the object (6 kg)
- ( v ) is the velocity of the object (4 m/s)

[ KE = \frac{1}{2} \times 6 \times 4^2 = 48 \text{ J} ]

The potential energy stored in the spring when compressed is given by: [ PE = \frac{1}{2}kx^2 ]

Where:

- ( k ) is the spring constant (12 kg/s²)
- ( x ) is the compression distance (unknown)

Setting the kinetic energy equal to the potential energy: [ 48 = \frac{1}{2} \times 12 \times x^2 ] [ 48 = 6x^2 ] [ x^2 = \frac{48}{6} = 8 ] [ x = \sqrt{8} \approx 2.83 \text{ meters} ]

Therefore, the spring will compress approximately 2.83 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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