Vector:
𝑣⃗ = 𝑣𝑥 𝑙̂ + 𝑣𝑦 𝐽̂
Magnitude:
|𝑣⃗| = 𝑣 = √𝑣𝑥2 + 𝑣𝑦2
Uniform Circular Motion
𝑅
𝑎av =
Average acceleration:
𝑣𝑦
𝜃 = tan−1 ( )
Direction:
𝑣
|𝛥𝑣⃗| = 1 𝛥𝑠
Velocity:
𝑣𝑥
|𝛥𝑣
⃗⃗|
𝛥𝑡
𝑣 𝛥𝑠
= 1
𝑅 𝛥𝑡
𝑣1 𝛥𝑠
Instantaneous acceleration: 𝑎 = lim
∆𝑡→0 𝑅 𝛥𝑡
𝑣av-𝑥 =
Average velocity:
∆𝑥
=
∆𝑡
Instantaneous velocity:
𝑣𝑥 = lim
Average acceleration:
𝑎av−𝑥 =
∆𝑥
∆𝑡→0 ∆𝑡
∆𝑣
∆𝑡
𝑥2 −𝑥1
2𝜋𝑅
Tangential velocity:
𝑣=
=
Centripetal acceleration:
𝑎rad =
Centripetal force:
𝐹net = 𝑚𝑎rad = 𝑚
𝑑𝑡
𝑣 −𝑣1
= 2
𝑇=
𝑇
𝑡2 −𝑡1
𝑑𝑥
𝑣
= 1 lim
𝑣2
𝑅
=
𝛥𝑠
𝑅 ∆𝑡→0 𝛥𝑡
𝑡
𝑟ⅇ𝑣
4𝜋2 𝑅
𝑇2
𝑣2
𝑡2 −𝑡1
𝑅
Instantaneous acceleration:
Non-uniform circular motion:
∆𝑣𝑥 𝑑𝑣𝑥 𝑑2 𝑥
=
= 2
∆𝑡→0 ∆𝑡
𝑑𝑡
𝑑𝑡
𝑎𝑥 = lim
Constant velocity:
𝑣2
𝑅
𝑑|𝑣⃗|
𝑎tan =
𝑅
𝑎rad =
𝑥 = 𝑣𝑡
Constant acceleration equation
1
𝑥 = 𝑥0 + 𝑣0𝑥 𝑡 + 𝑎𝑥 𝑡 2
2
1
𝑥 − 𝑥0 = (𝑣0𝑥 + 𝑣𝑥 )𝑡
2
𝑣𝑥 = 𝑣0𝑥 + 𝑎𝑥 𝑡
Relative Velocity:
𝑣𝑃∕𝐴−𝑥 = 𝑣𝑃/𝐵−𝑥 + 𝑣𝐵/𝐴−𝑥
𝑣𝐴∕𝐵−𝑥 = −𝑣𝐵/𝐴−𝑥
Two or Three Dimension
2
𝑣𝑥2 = 𝑣0𝑥
+ 2𝑎𝑥 (𝑥 − 𝑥0 )
Free fall
𝑎𝑦 = −𝑔 = −9.8 m/s2
1
ℎ = 𝛥𝑦 = 𝑣𝑜𝑦 𝑡 − 𝑔𝑡 2
2
1
ℎ = 𝛥𝑦 = (𝑣0𝑦 + 𝑣𝑦 )𝑡
2
𝑣𝑦 = 𝑣0𝑦 − 𝑔𝑡
Vector:
𝑣⃗𝑃∕𝐴 = 𝑣⃗𝑃/𝐵 + 𝑣⃗𝐵/𝐴
Magnitude:
2
2
|𝑣⃗𝑃∕𝐴 | = 𝑣 = √𝑣𝑃/𝐵
+ 𝑣𝐵/𝐴
Direction:
𝜃 = tan−1 (
Newton’s First Law:
𝑣𝑃/𝐵
𝑣𝐵/𝐴
)
∑𝐹⃗ = 0,
∑𝐹𝑥 = 0
Newton’s Second Law: ∑𝐹⃗ = 𝑚𝑎⃗, ∑𝐹𝑥 = 𝑚𝑎⃗ ∑𝐹𝑦 = 𝑚𝑎⃗
Newton’s Third Law:
2
𝑣𝑦2 = 𝑣0𝑦
− 2𝑔(𝛥𝑦)
Velocity and position by integration
𝐹⃗𝐴 𝑜𝑛 𝐵 = −𝐹⃗𝐵 𝑜𝑛 𝐴
Weight:
𝑤
⃗⃗⃗ = 𝑚𝑔⃗
Kinetic friction force:
𝑓𝑘 = 𝜇𝑘 𝑛
Static friction force:
𝑓𝑠 ≤ (𝑓𝑠 )max = 𝜇𝑠 𝑛
𝑡
𝑣𝑥 = 𝑣𝑜𝑥 + ∫ 𝑎𝑥 𝑑𝑡
0
𝑡
𝑥 = 𝑥𝑜 + ∫ 𝑣𝑥 𝑑𝑡
0
Projectile Motion
Horizontal
Vertical
𝑎𝑥 = 0
𝑎𝑦 = −𝑔 = −9.8 𝑚/𝑠 2
𝑥 = 𝑥0 + 𝑣0𝑥 𝑡
𝑦 = 𝑦0 + 𝑣𝑜𝑦 𝑡 − 𝑔𝑡 2
𝑣𝑥 = 𝑣0𝑥
𝑣𝑦 = 𝑣0𝑦 − 𝑔𝑡
𝑣𝑜𝑥 = 𝑣0 cos 𝛼𝜃0
𝑣𝑜𝑦 = 𝑣0 sin 𝜃0
1
2
1
𝑥 = (𝑣0 cos 𝜃0 ) 𝑡
𝑦 = (𝑣0 sin 𝜃0 )𝑡 − 𝑔𝑡 2
𝑣𝑥 = 𝑣0 cos 𝜃0
𝑣𝑦 = 𝑣0 sin 𝜃0 − 𝑔𝑡
2
𝑦 = (tan 𝜃0 )𝑥 −
9
2𝑣02 cos2 𝜃0
∑𝐹𝑦 = 0