Algorithm for Projectile Problems:
Always start with
a
ØHtL= < 0, -g>
, where
g=32 ft
ÅÅÅÅÅÅÅÅÅÅÅ
sec2
or
9.8 m
ÅÅÅÅÅÅÅÅÅÅÅ
sec2
(or g(t) if gravity is not constant, e.g. on planet Z)
1. Write down the "Initial Conditions" for
v
ØHtL
and
r
ØHtL
, (i.e
v
ØH0L= < vo cosq,vo sinq >
where q is in rad and
r
ØH0L= < 0, height off the ground >
)
2. Integrate
a
ØHtL
to find
v
ØHtL
using the initial condition
v
ØH0L= < vo cosq,vo sinq >
3. Integrate
v
ØHtL
to get
r
ØHtL
using the initial condition
r
ØH0L= < 0, height off the ground >
4. Answer the actual question, frequently solving for t first
Note:
To find range,
xHtL
: set the height,
y
component of
r
ØHtL
, equal to zero, solve for t and insert it in the x component of
r
ØHtL
To find maximum height: set
vyHtL
, the
y
component of
v
ØHtL
, equal to zero, solve for t and insert it in the y component of
r
ØHtL
To find its speed at impact: set the height
yHtL
equal to zero to find the time of impact, insert this t in both components of
v
ØHtL
, and compute
»»v
ØHtL»»
Note: Check meters vs. feet. Check radians vs. degrees
Arc Length:
Arc Length =
Ÿt1
t2»»v
ØHtL»»„t
Tangental and Normal Components of Acceleration:
aTHtL=d»»v
ØHtL»»
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
dt
aNHtL= k I»»v
ØHtL»»M2
or
aNHtL=
»»v
ØHtL x a
ØHtL»»
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
»»v
ØHtL»»
or
aNHtL
=
$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
I»»a
ØHtL»»M2-aT2
Curvature:
k =
aN
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
I»»v
ØHtL»»M2
or
k =
»»v
ØHtL x a
ØHtL»»
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
I»»v
ØHtL»»M3
k = 0 Ø straight line
k = const. Ø k =
1
ÅÅÅÅÅ
R
(
acircular =v2
ÅÅÅÅÅÅÅ
R
)
Think about:
a
Ø=aTT
`+aN N
`
=d»»v
ØHtL»»
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
dt T
`+ k I»»v
ØHtL»»M2 N
`
T
`
= const.
v
Ø
so
N
`
in perpendicular to
v
Ø
Exam III Formula Sheet.nb 1