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Energy transformation is an integral part of physics and applied math. Objects moving from one location to another are one of the most common ways these problems are asked, making a skier moving down a slope an obvious choice.

**As a skier moves down a slope, there is a transformation of energy between the skier’s potential and kinetic energy. The sum of energies must be equal to the sum of the energy at another point on the slope. There is only potential energy if the skier is static.**

## The Math

There is a transformation of energy between the different types of energy of the skier as they glide down the slope. The most important principle for this problem is the conservation of energy. The formula for this is shown here:

**K1+ U1= K2+ U2**

This formula states that the combined kinetic and potential energy of an object must be equal to the combined energy even after a change in height or velocity. There is also frictional energy involved in the system, but for simplicity, we will ignore that for this problem.

The equations for kinetic and potential energy can be broken down and shown like this:

**K = 1/2 x mv ^{2} and U =mgh**

For anyone not sure what the variables are, **K** stands for kinetic energy and **U** for potential energy. The other variables and constants represent:

- The
**m**is the mass of the object. - The
**v**represents the object’s velocity. - The
**g**represents gravitational the acceleration constant (9.81 m/s^{2}) - The
**h**is the height of the object from the datum.

Because the object’s mass and gravitation acceleration don’t change, all the energy transfer comes from a change in relative height and the velocity of the object.

As the skier goes down the slope, their velocity changes from zero, causing kinetic energy to change from zero to a positive value. Their height (or altitude) will also decrease and cause a decrease in potential energy.

## What Form of Energy Does a Skier Have at the Top of a Slope?

At the top of the slope, we assume the skier is not moving at all. This means velocity is zero, making the kinetic energy also equal to zero. All of the energy of the skier is from the potential energy.

Obviously, the mass of the skier can change from person to person, but once a value is known, that mass stays constant.

The height, **h**, is where all the energy is coming from. A very important part of the conservation of energy is selecting an appropriate datum that accurately represents the height of the skier.

For a problem like this, the datum is usually best set at the end of the slope, giving the height as the entire slope.

All of this means at a given height on the slope, the formula will be:

**U1= K2+ U2**

This is true for any point of the slope where the skier is moving.

## Let’s Recap

A person skiing down a slope is subject to a transformation of energy and this change is governed by the principle of conservation of energy. As the skier’s height from the bottom of the slope decrease, their potential energy decreases.

However, their total energy must remain constant for that system. The result is the velocity of the skier is determined by the height left on the slope.

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