Derivation of Kinematic
Equations
Sama Ehab, Mirna Amr, Nancy Mohamed, Salma Ammar
𝒗𝒊 + 𝒗𝒇
𝜟𝒙 = (
)𝜟𝒕
𝟐
Δ𝑥
•
Average velocity is represented as: 𝑣 =
•
Average velocity can also be represented as: 𝑣 =
•
Rearrange the equation above to solve for Δ𝑥
Δ𝑡
𝑣𝑖 +𝑣𝑓
2
Δ𝑥 = 𝑣Δ𝑡
•
𝑣𝑖 +𝑣𝑓
Substitute 𝑣 with
𝑣𝑖 + 𝑣𝑓
Δ𝑥 = (
)Δ𝑡
2
2
in the equation above to get our first kinematic equation.
𝛥𝑥 =
1
𝑏𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡 + (𝑙𝑒𝑛𝑔𝑡ℎ × 𝑤𝑖𝑑𝑡ℎ)
2
𝛥𝑥 =
1
𝛥𝑡 𝑣𝑓 − 𝑣𝑖 + 𝑣𝑖 × 𝛥𝑡
2
1
1
= 𝑣𝑓 𝛥𝑡 − 𝑣𝑖 𝛥𝑡 + 𝑣𝑖 𝛥𝑡
2
2
1
1
= 𝑣𝑓 𝛥𝑡 + 𝑣𝑖 𝛥𝑡
2
2
Δ𝑥 = (
o
We know that the area under the graph is
the displacement (Δ𝑥).
𝑣𝑓 + 𝑣𝑖
)𝛥𝑡
2
𝒗𝒇 = 𝒗𝒊 + 𝒂𝜟𝒕
Δ𝑣
Δ𝑡
•
Acceleration is represented as: 𝑎 =
•
Substitute Δ𝑣 with 𝑣𝑓 − 𝑣𝑖
𝑣𝑓 − 𝑣𝑖
𝑎=
Δ𝑡
•
Rearrange the equation above to solve for 𝑣𝑓
𝑣𝑓 = 𝑣𝑖 + 𝑎Δ𝑡
𝑣𝑓 = 𝐵𝐶
𝑣𝑓 = 𝐵𝐷 + 𝐷𝐶
𝑣𝑓 = 𝐵𝐷 + 𝑣𝑖
𝑎=
𝐵𝐷
𝐴𝐷
𝑎=
𝐵𝐷
Δ𝑡
𝐵𝐷 = 𝑎Δ𝑡
𝑣𝑓 = 𝑣𝑖 + 𝑎Δ𝑡
𝟏
𝜟𝒙 = 𝒗𝒊 𝜟𝒕 + 𝒂𝜟𝒕𝟐
𝟐
𝑣𝑖 +𝑣𝑓
•
Start with the first kinematic equation (XVT) Δ𝑥 =
•
Substitute 𝑣𝑓 with our second kinematic equation (VAT) 𝑣𝑓 = 𝑣𝑖 + 𝑎Δ𝑡.
Δ𝑥 =
𝑣𝑖 + (𝑣𝑖 + 𝑎Δ𝑡)
Δ𝑡
2
2𝑣𝑖 𝑎Δ𝑡
+
Δ𝑡
2
2
•
Expand the equation
•
Simplify the equation Δ𝑥 = 𝑣𝑖 Δ𝑡 + 𝑎Δ𝑡 2
Δ𝑥 =
1
2
2
Δ𝑡.
𝛥𝑥 =
1
𝑏𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡 + (𝑙𝑒𝑛𝑔𝑡ℎ × ℎ𝑒𝑖𝑔ℎ𝑡)
2
=
1
Δ𝑡 𝐵𝐷 + (𝑣𝑖 × Δ𝑡)
2
𝑎=
𝐵𝐷
𝐴𝐷
𝑎=
𝐵𝐷
Δ𝑡
𝐵𝐷 = 𝑎Δ𝑡
=
1
Δ𝑡 𝑎Δ𝑡 + 𝑣𝑖 Δ𝑡
2
1
𝛥𝑥 = 𝑣𝑖 𝛥𝑡 + 𝑎𝛥𝑡 2
2
𝒗𝟐𝒇 = 𝒗𝟐𝒊 + 𝟐𝒂𝜟𝒙
•
𝑣𝑖 +𝑣𝑓
Start with the first kinematic equation (XVT) Δ𝑥 = (
•
2
)Δ𝑡.
Use the second kinematic equation (VAT) 𝑣𝑓 = 𝑣𝑖 + 𝑎Δ𝑡,
and rearrange it to solve for t.
𝑣𝑓 − 𝑣𝑖
Δ𝑡 =
𝑎
𝑣𝑓 − 𝑣𝑖
• Substitute Δ𝑡 in our first equation with
.
𝑎
𝑣𝑖 +𝑣𝑓
Δ𝑥 = (
2
𝑣𝑓 − 𝑣𝑖
)(
𝑎
).
𝑣𝑓2 −𝑣𝑖2
•
Multiply the fractions and simplify Δ𝑥 = (
•
We can rearrange the equation above to solve for 𝑣𝑓2
𝑣𝑓2 = 𝑣𝑖2 + 2𝑎Δ𝑥
2𝑎
).
1
Δ𝑥 = (𝑏𝑎𝑠𝑒1 + 𝑏𝑎𝑠𝑒2)(ℎ𝑒𝑖𝑔ℎ𝑡)
2
1
Δ𝑥 = 𝑣𝑖 + 𝑣𝑓 Δ𝑡
2
𝑣𝑓 − 𝑣𝑖
𝑎=
Δ𝑡
Δ𝑡 =
=
𝑣𝑓 − 𝑣𝑖
𝑎
1
𝑣 + 𝑣𝑓
2 𝑖
𝑣𝑓 − 𝑣𝑖
𝑎
1 𝑣𝑓2 − 𝑣𝑖2
=
2
𝑎
2𝑎Δ𝑥 = 𝑣𝑓2 − 𝑣𝑖2
𝑣𝑓2 = 𝑣𝑖2 + 2𝑎Δ𝑥
Resources
“Derivation of Equations of Motion.” Byju’s, 28 June 2018,
https://byjus.com/physics/derivation-of-equation-of-motion/#Derivation-of-ThirdEquation-of-Motion.
“Kinematic Equations.” Pasco, https://www.pasco.com/products/guides/kinematic-equations.
“What are the kinematic formulas?” Khan Academy,
https://www.khanacademy.org/science/physics/one-dimensional-motion/kinematicformulas/a/what-are-the-kinematic-formulas.
“Derivation of Kinematic Equations.”
https://www.muncysd.org/site/handlers/filedownload.ashx?moduleinstanceid=2437&data
id=4035&FileName=Kinematic%20Eqns.pdf.