3 how fast can things go?
3.1 motion
| name | symbol | units |
|---|---|---|
| final velocity | \(v\) | \((ms^{-1})\) |
| initial velocity | \(u\) | \((ms^{-1})\) |
| acceleration | \(a\) | \((ms^{-2})\) |
| time | \(t\) | \((s)\) |
| displacement | \(s\) | \((m)\) |
| centripetal acceleration | \(a_c\) | \((ms^{-2})\) |
| centripetal force | \(F_c\) | \((N)\) |
| mass | \(m\) | \((kg)\) |
| velocity | \(v\) | \((ms^{-1})\) |
| distance | \(r\) | \((m)\) |
| momentum | \(p\) | \((kgms^{-1})\) |
3.1.1 newton’s three laws of motion
- every object in a state of uniform motion will remain in that state of motion unless an external force acts on it
- force equals mass times acceleration \(F=ma\)
- for every action there is an equal and opposite reaction
3.1.2 constant acceleration
\[v=u+at\] \[v^2=u^2+2as\] \[s=\frac{1}{2}(u+v)t\] \[s=ut+\frac{1}{2}at^2\] \[s=vt-\frac{1}{2}at^2\]
3.1.3 circular motion
\[a_c=\frac{v^2}{r}\]
\[F_c=\frac{mv^2}{r}\]
3.1.4 momentum
\[P=mv\]
conservation of momentum states that total momentum before and after a collision is equal
\[\sum{p_\text{initial}}=\sum{p_\text{final}}\]
3.2 energy
energy is a measure of the ability of something to cause a change in the world and its unit is joules \((J)\)
| name | symbol | units |
|---|---|---|
| kinetic energy | \(E_k\) | \((J)\) |
| strain energy | \(E_s\) | \((J)\) |
| gravitational potential energy | \(E_g\) | \((J)\) |
| mass | \(m\) | \((m)\) |
| velocity | \(v\) | \((ms^{-1})\) |
| spring constant | \(k\) | \((Nm^{-1})\) |
| displacement | \(x\) | \((m)\) |
| gravitational field strength | \(g\) | \((ms^{-2})\) |
| height above surface | \(h\) | \((m)\) |
\[E_k=\frac{1}{2}mv^2\] \[E_s=\frac{1}{2}kx^2\] \[E_g=mg\Delta{}h\]
3.2.1 conservation of energy
conservation of energy states that total energy of an isolated system remains constant
\[\sum{E_\text{initial}}=\sum{E_\text{final}}\]
3.3 special relativity
3.3.1 einsteins two postulates
- the laws of physics are the same in all inertial frames of reference
- the speed of light in free space has the same value \(c=3*10^8ms^{-1}\) in all inertial frames of reference
3.3.2 frame of reference
frame of reference is a set of physical reference points that uniquely fix (locate and orient) the coordinate system and standardize measurements
an inertial frame of reference is a reference frame that is not accelerating
3.3.3 lorentz factor
\[\gamma{}=\frac{1}{\sqrt{1-(\frac{v}{c})^2}}\] \[v=c\sqrt{1-(\frac{1}{\gamma})^2}\] as the velocity \(v\) approaches the speed of light \(c\), the lorentz factor \(\gamma\) approaches \(\infty\)
the \(x\) axis is scaled by a factor of \(10^8\) so we can see the effect
3.3.4 time dilation
proper time is the time interval between two events happening at the same location
\[t=t_0\gamma=\frac{t_0}{\sqrt{1-(\frac{v}{c})^2}}\] \[v=c\sqrt{1-(\frac{t_0}{t})^2}\]
3.3.5 length contraction
proper length is the length of an object as measured by an observer who is stationary relative to that object
\[L=\frac{L_0}{\gamma}=L_0\sqrt{1-(\frac{v}{c})^2}\] \[v=c\sqrt{1-(\frac{L}{L_0})^2}\]
3.3.6 mass energy
\[E_0=mc^2\] \[E_\text{total}=E_0+E_k=\gamma{}mc^2\] \[E_k=(\gamma{}-1)mc^2\]