# A spring with a constant of #6# #kgs^-2# is lying on the ground with one end attached to a wall. An object with a mass of #7# # kg# and speed of #2# #ms^-1# collides with and compresses the spring until it stops moving. How much will the spring compress?

When the initial kinetic energy,

The mass in motion will transform its kinetic energy into spring potential energy.

The initial energy of motion:

Last spring potential energy:

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To find the compression of the spring, we can use the principle of conservation of mechanical energy, which states that the total mechanical energy of a system remains constant if only conservative forces act on it. Initially, the object has kinetic energy, which is converted into potential energy stored in the compressed spring.

The formula for the potential energy stored in a compressed spring is:

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

where:

- ( PE ) is the potential energy stored in the spring,
- ( k ) is the spring constant, and
- ( x ) is the compression of the spring.

The kinetic energy of the object initially is:

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

where:

- ( m ) is the mass of the object, and
- ( v ) is the initial velocity of the object.

Since the object comes to rest after compressing the spring, all of its initial kinetic energy is converted into potential energy stored in the spring.

Setting the initial kinetic energy equal to the potential energy of the spring:

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

Substituting the given values:

- ( m = 7 , \text{kg} ) (mass of the object)
- ( v = 2 , \text{m/s} ) (initial velocity of the object)
- ( k = 6 , \text{kg/s}^2 ) (spring constant)

[ \frac{1}{2} \times 7 \times (2)^2 = \frac{1}{2} \times 6 \times x^2 ]

[ 7 \times 4 = 6x^2 ]

[ 28 = 6x^2 ]

[ x^2 = \frac{28}{6} ]

[ x^2 = \frac{14}{3} ]

[ x = \sqrt{\frac{14}{3}} ]

Thus, the spring compresses by approximately ( \sqrt{\frac{14}{3}} ) 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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