PX155 - C1 - work and kinetic energy

• from this equation

• aside:

so,

• if it is a particle of mass $m$:

where, $T=\frac{1}{2}m{v}^2$

• $\int F(x)dx=\mathrm{\Delta }T$ is work done $W$
• if $F$ is constant, so not a function of position,$$W=\int_{x_A}^{x_{B}}Fdx= F(x_B-x_A)$$

• however, often in infinitesimal step where $F(x)$ can be considered constant

work in 3D

• above generalises to $dW=F_{x}dx+F_{y}dy+F_{z}dz$
  • $\overrightarrow{F}=(F_x,F_y,F_z)$
  • $dW=\overrightarrow{F}\cdot d\overrightarrow{r}$
    • $\overrightarrow{d}r=(dx,dy,dz)$
    • in x-direction: $F_{x}dx=\frac{m}{2}d(v_2^2)$, similar for y and z
    • therefore,

- so,

- so in 3D, $T=\frac{1}{m}\overrightarrow{v}\cdot \overrightarrow{v}=\frac{1}{2}m|\overrightarrow{v}{|}^2=\frac{1}{2}m{v}^2$
- where, $v=|\overrightarrow{v}|$
- and,

- work done by $\overrightarrow{F}$ along a "slice" of the path $d\overrightarrow{r}$:

- $dW$ is the position of $\overrightarrow{F}$ along the path times an infinitesimal distance moved during that path