# Physics equations/Work, energy, power

#### Define *kinetic energy*, *work*, and *potential energy*

[edit | edit source]Symbols used for kinetic energy include *KE* (or *K.E.*), *T*, and *K* . Using the latter, we have for ** kinetic energy**:

In **shorthand-to-powerful notation**, we can write these expressions for
**work**:

Potential energy can be defined for conservative forces. Two important conservative forces are gravity and that of an ideal spring. Symbols for potential energy include *PE* (or *PE*) and *V*, and *U*:

At Earth's surface, the gravitational P.E. is *U_g = mgy*, where *g* = 9.8 m/s^{2} ("standard gravity"). and for a spring, *U = ½kx ^{2}*, where

*x*is the displacement from the spring's equilibrium position. The total mechanical energy (often written as M.E. or just E) is conserved if all forces are conservative: Mechanical energy is not conserved if non-conservative forces are present. Friction is an important non-conservative force. Energy conservation can be stated as:

***Problem:** Describe a conservative and a nonconservative force in sports.

[edit | edit source]**Solution:**

******Problem:** Use calculus to show why these equations are true

[edit | edit source]**Solution:** The work done accelerating a particle during the infinitesimal time interval *dt* is given by the dot product of *force* and *displacement*:

Applying the product rule we see that:

Therefore (assuming constant mass so that *dm*=0), the following can be seen:

Since this is a total differential (that is, it only depends on the final state, not how the particle got there), we can integrate it and call the result kinetic energy:

Thus the change in kinetic energy equals the integral of the net force over distance.^{[1]}

***Problem:** Energy conservation with a mass, a spring, and a hill

[edit | edit source]A spring with spring constant, k, is compressed by a distance x_{0}. A mass, m, is placed on the spring and it is released, sliding initially along a horizontal surface at a speed of v_{1}. Sliding without friction, it encounters a hill where it rises to a height, h_{2} as it continues to move at a speed of v_{2}. Finally it reaches a height of h_{3}, where it stops. Write equations to model the particle's speed and height at each stage.

**Solution:** *½kL ^{2} = ½mv_{0}^{2} = ½mv_{1}^{2} + mgh_{1} = mgh_{2}*

#### Power

[edit | edit source]Power, P, is the rate at which work is done or energy is consumed. The SI unit of power is Watt (W) which equals one Joule per second (J/s).

****Problem:** Show that power is the dot product of force and velocity

[edit | edit source]**Solution:** This solution was extracted from Physics with Calculus (Wikibook) and uses the notation of that chapter:

Power is the rate of doing work. To come up with a useful expression for this, consider a short amount of time . How much work is done in that time? Well, by the definition of power, it is very nearly . By the definition of work, this is where is the displacement which occurs in . We have

So,

,

which in the limit as goes to zero (which is when our "equations" become exact),

.

Equivalently, we could have gotten the expression by simply differentiating our expression for work. No matter the derivation, because it simply does not matter that much; we have a useful expression for power.