Summary ALL-IN-ONE Formula Sheet for PAT & ENGAA

PAT Formula Sheet for Final Round Revision




Songyue Wang

, PAT Formula Sheet for Final Round Revision

by Songyue Wang

1 Mechanics

1.1 Constant Acceleration Motion
1.1.1 SUVAT equations

𝑣𝑣 = 𝑢𝑢 + 𝑎𝑎𝑎𝑎
(𝑢𝑢 + 𝑣𝑣)𝑡𝑡
𝑠𝑠 =
2
𝑎𝑎𝑡𝑡 2
𝑠𝑠 = 𝑢𝑢𝑢𝑢 +
2
𝑎𝑎𝑡𝑡 2
𝑠𝑠 = 𝑣𝑣𝑣𝑣 −
2
𝑣𝑣 2 = 𝑢𝑢2 + 2𝑎𝑎𝑎𝑎
1.1.2 Relationship between acceleration, velocity and displacement
𝑑𝑑𝑠𝑠
= 𝑣𝑣
𝑑𝑑𝑡𝑡
𝑑𝑑𝑣𝑣
= 𝑎𝑎
𝑑𝑑𝑡𝑡
∫ 𝑎𝑎 𝑑𝑑𝑡𝑡 = 𝑣𝑣

∫ 𝑣𝑣 𝑑𝑑𝑡𝑡 = 𝑠𝑠
1.1.3 Newton’s 2nd law

𝐹𝐹 = 𝑚𝑚𝑚𝑚
𝑑𝑑𝑣𝑣
𝐹𝐹 = 𝑚𝑚
𝑑𝑑𝑡𝑡
1.1.4 Projectile motion

gx 2
=y x tan θ −
2V 2 cos 2 θ
1.2 Resistive force
1.2.1 Proportional model of resistive force

f = − kv

1.2.2 Air resistance

1
f = c ρ Av 2
2

, c: drag coefficient

1.2.3 Terminal speed

mg
vT = (proportional model)
k

2mg
vT = (air resistance model)
cρ A

1.3 Spring
1.3.1 Hooke’s law

𝐹𝐹 = −𝑘𝑘𝑘𝑘
1.3.2 Energy in a spring
1
𝐸𝐸𝑠𝑠 = 𝑘𝑘𝑥𝑥 2
2
1.3.3 Combination of springs
1 1 1
= + (𝑖𝑖𝑖𝑖 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠)
𝑘𝑘 𝑇𝑇 𝑘𝑘1 𝑘𝑘2
𝑘𝑘 𝑇𝑇 = 𝑘𝑘1 + 𝑘𝑘2 (𝑖𝑖𝑖𝑖 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝)
1.4 Momentum

𝑝𝑝 = 𝑚𝑚𝑚𝑚
1.5 Impulse

I= m∆v
dp ∆v
=
F = m
dt t
1.6 Linear collision
1.6.1 Conservation of linear momentum

𝑚𝑚1 𝑢𝑢1 + 𝑚𝑚2 𝑢𝑢2 + ⋯ + 𝑚𝑚𝑛𝑛 𝑢𝑢𝑛𝑛 = 𝑚𝑚1 𝑣𝑣1 + 𝑚𝑚2 𝑣𝑣2 + ⋯ + 𝑚𝑚𝑛𝑛 𝑣𝑣𝑛𝑛
1.6.2 Coefficient of restitution
𝑣𝑣𝑟𝑟 𝑣𝑣2 − 𝑣𝑣1
𝑒𝑒 = =
𝑢𝑢𝑟𝑟 𝑢𝑢1 − 𝑢𝑢2
1.7 Energetics
1.7.1 Kinetic energy
1
𝐸𝐸𝑘𝑘 = 𝑚𝑚𝑣𝑣 2
2