# A ball with a mass of #650 g# is projected vertically by a spring loaded contraption. The spring in the contraption has a spring constant of #32 (kg)/s^2# and was compressed by #7/8 m# when the ball was released. How high will the ball go?

The height is

The spring constant is

The compression is

The potential energy in the spring is

This potential energy will be converted to kinetic energy when the spring is released and to potential energy of the ball

Let the height of the ball be

Then ,

The potential energy of the ball is

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To find the maximum height the ball will reach, you can use the conservation of mechanical energy. The potential energy stored in the spring when compressed is equal to the kinetic energy of the ball when it is released. This can be expressed as:

[PE_{spring} = KE_{ball}]

[PE_{spring} = \frac{1}{2} k x^2]

[KE_{ball} = \frac{1}{2} m v^2]

Since the ball is projected vertically, its velocity at maximum height is 0. Therefore, the kinetic energy at maximum height is 0.

[PE_{spring} = \frac{1}{2} k x^2]

[mgh = \frac{1}{2} k x^2]

[h = \frac{kx^2}{2mg}]

Substitute the given values:

[h = \frac{(32)(\frac{7}{8})^2}{2(0.65)(9.8)}]

[h ≈ 0.218,m]

So, the ball will go approximately 0.218 meters high.

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The ball will reach a maximum height of approximately 1.79 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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