# A spring with a constant of #7 (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 #8 m/s# collides with and compresses the spring until it stops moving. How much will the spring compress?

This question relates to energy conservation, as you have already observed.

In the spring, all of the object's kinetic energy will transform into elastic potential energy.

This indicates that the amount of force needed to compress or stretch the spring depends on how much of it has already been done so.

In terms of math, we write,

The force will always oppose the spring's displacement because the spring always seeks to return to its equilibrium length, which is the source of the negative sign.

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To find the compression of the spring, you can use the conservation of mechanical energy. The initial kinetic energy of the object will be converted into the potential energy stored in the compressed spring. The equation to use is:

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

where:

- (m) is the mass of the object (3 kg)
- (v) is the velocity of the object (8 m/s)
- (k) is the spring constant (7 kg/s^2)
- (x) is the compression of the spring (unknown)

Rearranging the equation to solve for (x):

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

Substituting the given values:

[x = \sqrt{\frac{(3 \text{ kg}) \times (8 \text{ m/s})^2}{7 \text{ kg/s}^2}}]

[x ≈ \sqrt{\frac{192}{7}}]

[x ≈ \sqrt{27.42}]

[x ≈ 5.23 \text{ meters}]

Therefore, the spring will compress approximately 5.23 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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