Kinetic and potential energy
The roller-coaster swap
- At the top of a drop a coaster crawls; at the bottom it races.
- Height has turned into speed — potential energy into kinetic energy.
- Energy is just moving between two stores.
Gravitational potential energy
- Lifting a mass $m$ through a height $\Delta h$ stores $\Delta E_{\text{P}} = mg\Delta h$.
- It comes from the work done against gravity: $W = mg \times \Delta h$.
A $2.0\ \text{kg}$ book is lifted $5.0\ \text{m}$. How much gravitational PE does it gain? (Use $g = 9.81\ \dfrac{\text{m}}{\text{s}^2}$.)
$\Delta E_{\text{P}} = mg\Delta h = 2.0 \times 9.81 \times 5.0 \approx 98\ \text{J}$.
Kinetic energy
- A mass $m$ moving at speed $v$ has $E_{\text{K}} = \tfrac{1}{2}mv^{2}$.
- It comes from the work done to speed it up: $W = Fs = \tfrac{1}{2}mv^{2}$.
A $4.0\ \text{kg}$ trolley moves at $3.0\ \dfrac{\text{m}}{\text{s}}$. What is its kinetic energy?
$E_{\text{K}} = \tfrac{1}{2}mv^{2} = \tfrac{1}{2} \times 4.0 \times 3.0^{2} = 18\ \text{J}$.
The energy swap
- On a frictionless ramp, GPE becomes KE: $mgh = \tfrac{1}{2}mv^{2}$, so $v = \sqrt{2gh}$.
- With friction, some of it becomes thermal energy instead.
As a ball rolls down a frictionless ramp, its gravitational PE turns mainly into:
With no friction, all the lost GPE becomes kinetic energy: $mgh = \tfrac{1}{2}mv^{2}$.
Energy and momentum
- Combining $p = mv$ with $E_{\text{K}} = \tfrac{1}{2}mv^{2}$ gives $E_{\text{K}} = \dfrac{p^{2}}{2m}$.
- Handy when you know the momentum but not the speed.
Kinetic energy written in terms of momentum $p$ is:
From $p = mv$ and $E_{\text{K}} = \tfrac{1}{2}mv^{2}$, substituting $v = p/m$ gives $E_{\text{K}} = \dfrac{p^{2}}{2m}$.
Energy methods
- Write the energy at the start and end, then balance the books.
- Spring: a block's KE becomes elastic PE $\tfrac{1}{2}kx^{2}$ at greatest compression.
- Bounce: the height ratio $\dfrac{h_2}{h_1}$ is the fraction of energy kept.
A block slides into a spring on a frictionless surface. At greatest compression its kinetic energy has all become ____ potential energy.
KE → elastic PE $= \tfrac{1}{2}kx^{2}$ at the point of greatest compression, then back to KE as the spring pushes it off.
When friction acts, some mechanical energy becomes thermal energy.
Yes — that "lost" energy is not destroyed; it heats the surfaces and surroundings.
You've got it
- gravitational PE $\Delta E_{\text{P}} = mg\Delta h$; kinetic energy $E_{\text{K}} = \tfrac{1}{2}mv^{2}$
- on a frictionless ramp $mgh = \tfrac{1}{2}mv^{2}$, so $v = \sqrt{2gh}$
- in momentum terms, $E_{\text{K}} = \dfrac{p^{2}}{2m}$