PX155 - C4 - using energy conservation

• if no mention of time in a problem and only conservative forces act, try using energy conservation
• eg: object released from rest at a height $h=3000km$ above the earth
  • ignore air resistance
  • to find: speed at which it hits the ground
    • $E=T+U$
    • when released:
      • $U=-\frac{G{M}_{E}m}{R_E+h}$
      • $T=0$
    • when it hits the ground:
      • $U=-\frac{G{M}_{E}m}{R_E}$
      • $T=\frac{1}{2}m{v}^2$
    • equating

- plugging in the numbers: $v=6.3km{s}^{-1}$
- if $U=mgh$, $v=7.7km{s}^{-1}$

• eg: escape velocity : what speed is needed to escape the gravitational pull of a body of mass $M$ and radius $R$?
  • on the surface:
    • $T=\frac{1}{2}m{v}_E^2$
    • $U=-\frac{GMm}{R}$
  • at $r=\mathrm{\infty }$:
    • $T=0$
    • $U=0$
  • equating:

- plugging in numbers:
- for earth: $v_E=11.2km{s}^{-1}$
- for sun: $v_E=618km{s}^{-1}$