MATT SAYS, “HERE’S SOME USEFUL ENERGY EQUATIONS”
Relating force and energy:
Tackling Relativity:
"E
"p
= Faverage =
"x
"t
OR
From two equations spring everything :
Energy and Momentum
E particle = E particle' s + K particle' s = mc 2 + K(?)
dE
dp
= Finstantaneous =
dx
dt
rest
motion
there is NO good simple
expression for K in relativity so
mc 2
E particle =
!
v 2particle
1"
c2
Now,
mv particle
p particle =
v 2particle
1"
c2
These two equations combine for the
SUPER IMPORTANT EQUATION :
Work defined:
W = "(Mechanical Energy)
W = +F
•4
"r
1
42
3
if both are constant
OR
rf
W from ri to r f = + # F
(r2
) •4
dr
14
3
ri
E 2 " p 2c 2 = m 2c 4
if both are
constant
Note : work is measured in joules
but strictly speaking W $ E, but
rather W = "E.
!
You should also be able to use the energy and
momentum equations each to find velocity:
Need velocity?
v particle = ±c 1"
!
m 2c 4
2
E particle
OR
v particle =
!
pc
2
p " m 2c 2
Potential Energy (stored in a force field):
Potential Energy Examples
Type
U "
F
"U = - "W
so that
U = #F
•4
"r
1
42
3
if both are constant
gravity :
OR
#
GMm
GMm
"# 2
r
r
rf
U from ri to r f = # $ F
(r2
) •4
dr
14
3
ri
earth
if both are
constant
surface :
mgr " #mg
electric :
1 Qq
1 Qq
"
4 $%o r
4 $%o r 2
But when you need to create an expression for U(r),
then you must decide what ro corresponds to U(ro)=0.
rf
!
U(rf ) = " # F
(r2
) •4
dr
14
3
ro
if both are
constant
and the fundamental
theorem of calculus gives
dU(r )
F(r ) = "
dr
!
spring :
!
1 2
ks " #ks
2