Kinematic Relationships

Equations of Motion

As part of the Higher course, several Equations of Motion were considered. These formulae allowed the calculation of a range of variables relating to the motion of an object in Classical Mechanics. A deeper understanding of these formulae is required for the Advanced Higher course including where they came from...


v = u + at s = ut +1/2at2 v2 = u2 + 2as s = 1/2 ( u + v ) t


Each of the equations above has been derived from the same basic root - the Velocity, Displacement, Time formula :-


v = s/t


In order to derive the Equations of Motion, the use of Calculus is required. Calculus was created by Newton and Leibniz specifically to allow the mathematical description of change in numerical values, something that algebra couldn't do.


Displacement, Velocity and Acceleration

The above diagram shows the motion of an object travelling from Point O towards Point X.

The Object moves from O to P giving a displacement of 's' in time t

The Object moves from P to Q giving a displacement of 'Δs' in time Δt

Note - the Delta ( Δ ) symbol in the above diagram refers mathematically to the change in s or t.

The Average Velocity of the Object between points P and Q is therefore :-

Vav = Δs / Δt

The Instantaneous Velocity, however, is found using the following Calculus expression, where the value of Δt tends towards zero :-

This shows that by Calculus, Velocity can be found as the first differential of Displacement.

The Average Acceleration can be found through a similar method with a change in average Velocity (Δv) over a change in time (Δt) :-


aav = Δv / Δt


The Instantaneous Acceleration, however, is found using the following Calculus expression, where the value of Δt tends towards zero :-

This shows that by Calculus, Acceleration can be found as the first differential of Velocity. This also means that Acceleration can be found as the Second Differential of Displacement :-

Derivation of the Equations of Motion

The following calculations show the derivation of the Equations of Motion and are Examinable techniques :-


Equation 1 - v = u + at

Integration of Acceleration with respect to time :-

Equation 2 - s = ut + 1/2at2

Integration of Velocity with respect to time :-

Equation 3 - v2 = u2 +2as

Square both sides of EoM 1, then substitute in EoM 2 :-

Example 1 -

The displacement of an object over time can be described by the following equation :-

s(t) = 3.1t2 + 4.1t + 6

Calculate :-

The Velocity of the Object after 7 seconds

The Acceleration of the Object

Velocity after 7 Seconds :-

Acceleration of the Object :-