# Conservation of Energy

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The aerial acrobatics of the Russian Barre routine require exquisite balance, timing, and years of training. That's not enough though; conservation of energy plays a big role as well. At the top of the jump, Anna's energy is entirely potential but as she falls, her energy turns kinetic. Landing on the bar, the energy changes form once more, stored as elastic energy in the bending bar. This unit explores the basics of these three common forms of energy.

## Watch the Video

## Circus Physics: Conservation of Energy

Watch as Anna flips and twists in the air as part of the Russian Barre routine. Find out how conservation of energy helps make this spectacle possible.

### Questions to Consider While Watching the Video

- What forms of energy do you see?
- Where is the jumper's kinetic energy the greatest?
- Where is her potential energy greatest?
- How is she able to launch so high?
- Where does the needed energy come from?

## Digging Deeper

As Anna jumps and lands on the bar, her energy changes forms multiple times, but notice that her total energy, the bar on the right, never changes. This is because energy can change forms, but cannot be created or destroyed—in other words total energy in a system is conserved.

At the top of the jump, Anna's energy is entirely in the form of gravitational potential energy, *P*. *P* depends on Anna's height, *h*, above the ground, the acceleration due to gravity, *g*, and her mass, *m*:

*P* = *mgh*

As she begins to fall back down, her velocity increases as her height decreases. *P* decreases, but her energy of movement, called kinetic energy, *K*, increases. * K* depends only on Anna's mass, *m*, and velocity, *v*:

*K* = ½*mv*²

When Anna lands on the bar, her kinetic energy is transferred to bending the bar, and now takes the form of elastic energy, *U*. *U* depends on how deep the bar's bend is, *d*, and its "springiness", a constant *k*.

*U* = ½*kd*²

Notice that, even at the bottom of the bar's bend, Anna still has a tiny bit of potential energy.

## Your Turn

Use the concepts and formulas from this unit to figure out the following:

If Anna reaches a height of 5 meters, and she has a mass of 60 kg, how fast is she going when she leaves the bar?

##### Answer: ≈9.9 meters/second

The kinetic energy she has when she leaves the bar must be equal to the potential energy she has at the top of her flight. So set *P* = *m* x *g* x *h* equal to *K* = ½ x *m* x *v*². Since m appears on both sides of the equation, it will cancel out, meaning that the answer doesn't depend on how much she weighs! This leaves us with:

*g* x *h* = ½ x *v*²

Solving for v, inserting 9.8 m/s for g, and 5 meters for h, we get:

*v* = √(2 x 9.8 m/s x 5 m) ≈ 9.9 m/s)

## Further Explanation

If you'd like to learn more about conservation of energy and the Russian Barre, check out these links: