# 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 #3 m/s# collides with and compresses the spring until it stops moving. How much will the spring compress?

The spring will compress

We can either do this using conservation of energy or kinematics. I'll use conservation of energy.

Hopefully this helps!

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Quantities given:

We can calculate the kinetic energy of the object at the moment before it contacts the spring:

The potential energy of a spring is given as:

When the object comes to rest, it will be because the object has lost all of its kinetic energy and the energy has been transferred into the spring.

Substituting known values:

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To find the compression of the spring, we can use the principle of conservation of mechanical energy.

The initial kinetic energy of the object is given by:

( KE_{initial} = \frac{1}{2}mv^2 )

where: m = mass of the object (6 kg) v = initial velocity of the object (3 m/s)

( KE_{initial} = \frac{1}{2} * 6 , kg * (3 , m/s)^2 = 27 , J )

When the object compresses the spring, all of its kinetic energy is converted into potential energy stored in the spring. The potential energy stored in a spring is given by:

( PE_{spring} = \frac{1}{2}kx^2 )

where: k = spring constant (12 (kg)/s^2) x = compression of the spring (unknown)

Setting the initial kinetic energy equal to the potential energy stored in the spring:

( KE_{initial} = PE_{spring} )

( 27 , J = \frac{1}{2} * 12 , (kg)/s^2 * x^2 )

Solving for x:

( x^2 = \frac{27 , J}{\frac{1}{2} * 12 , (kg)/s^2} )

( x^2 = \frac{27 , J}{6 , (kg)/s^2} )

( x^2 = 4.5 , m^2 )

( x = \sqrt{4.5 , m^2} )

( x \approx 2.12 , m )

Therefore, the spring will compress approximately 2.12 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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