# Alfredo, who has a mass of 60 kilograms, jumped off a retaining wall. His kinetic energy when he landed on the ground was 822 joules. How high was the wall?

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## To calculate the height of the retaining wall, we can use the principle of conservation of energy. The potential energy of Alfredo when he was on the wall is converted into kinetic energy when he lands on the ground.

The formula to calculate potential energy is:

Potential Energy (PE) = mass (m) x gravity (g) x height (h)

The formula to calculate kinetic energy is:

Kinetic Energy (KE) = 0.5 x mass (m) x velocity (v)^2

Since Alfredo's mass remains constant, we can equate the potential energy before jumping off the wall to the kinetic energy when he lands on the ground:

m * g * h = 0.5 * m * v^2

Where:
m = 60 kilograms (mass of Alfredo)
g = 9.8 m/s^2 (acceleration due to gravity)
h = height of the wall (to be calculated)
v = velocity at which Alfredo lands on the ground

We are given that the kinetic energy (KE) is 822 joules, so we can substitute these values into the equation:

60 kg * 9.8 m/s^2 * h = 0.5 * 60 kg * v^2

Simplifying the equation further:

9.8 m/s^2 * h = 0.5 * v^2

Now, let's assume that Alfredo lands on the ground with zero initial velocity (v = 0), as he jumps off the retaining wall. In this case, the equation becomes:

9.8 m/s^2 * h = 0.5 * 0^2

This simplifies to:

9.8 m/s^2 * h = 0

Since the right side of the equation is zero, we can conclude that the height of the retaining wall (h) must also be zero. Therefore, the wall has zero height.

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## To determine the height of the retaining wall, we can use the conservation of mechanical energy, which states that the total mechanical energy of an object is conserved in the absence of non-conservative forces, such as friction or air resistance.

The total mechanical energy is the sum of the potential energy and kinetic energy of the object. When Alfredo jumps off the wall, he initially has gravitational potential energy, which gets converted to kinetic energy as he falls.

The formula for gravitational potential energy is:

Potential Energy = mass * gravity * height

where mass is the mass of the object (60 kg), gravity is the acceleration due to gravity (approximately 9.8 m/s² on Earth), and height is the vertical distance or height of the wall.

Given that the kinetic energy when Alfredo landed on the ground was 822 joules, we can equate the potential energy at the top of the wall to the kinetic energy at the bottom:

mass * gravity * height = kinetic energy

Substituting the given values:

60 kg * 9.8 m/s² * height = 822 J

Rearranging the equation to solve for height:

height = 822 J / (60 kg * 9.8 m/s²)

Calculating the height:

height = 1.39 meters

Therefore, the height of the retaining wall is approximately 1.39 meters.

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## m g h = Potential energy at top = 822

60 (9.81) h = 822

h = 1.4 meters

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