# 8.1: Energy and Force

**At Grade**Created by: CK-12

The law of *conservation of momentum* states that in any closed system (including the universe) the total quantity of momentum is constant. Momentum can be transferred from one body to another, but none is lost or gained. If a system has its momentum changed from the outside it is caused by an *impulse,* which transfers momentum from one body to another.

When any two bodies in the universe interact, they can exchange energy, momentum, or both. The law of *conservation of energy* states that in any closed system (including the universe) the total quantity of energy remains fixed. Energy is transferred from one form to another, but not lost or gained. If energy is put into a system from the outside or vice versa it is often in the form of *work,* which is a transfer of energy between bodies.

## Key Equations and Definitions

\begin{align*}\text{Energy}\begin{cases}
K = \tfrac{1}{2}mv^2 & \text{Kinetic energy (in Joules, kg m}^2 \text{s}^2 \text{)}\\
U_g = mgh & \text{Gravitational potential energy, J}\\
U_{\text{spring}} = \tfrac{1}{2} k({\Delta x})^2 & \text{Spring potential energy, J}
\end{cases}\end{align*}

\begin{align*}\text{Transfers}
\begin{cases}
W = F \cdot d & \text{Work is the dot product of force and displacement}\\
P = \frac{\Delta E}{\Delta t} & \text{Power is the rate of change of energy of a system, in Watts \ (J/s)}\\
J= \Delta p = F \Delta t & \text{Impulse is the change in a system's momentum}
\end{cases}\end{align*}

\begin{align*}\text{Conservation Laws}
\begin{cases}
\sum p_{\text{initial}} = \sum p_{\text{final}} & \text{Total momentum is constant in closed systems}\\
\sum E_{\text{initial}} = \sum E_{\text{final}} & \text{Total energy is constant in closed systems}\\
\sum K_{\text{initial}} = \sum K_{\text{final}} & \text{Kinetic energy conserved}\ \text{only}\ \text{in elastic collisions}\\
\end{cases}\end{align*}

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