Equations of Motion : Definition, Formulas, & FAQs

Introduction to Equations of Motion?

Physics equations of motion are a basic concept that we all might have studied in middle school. It is important to understand what it means and what we do with it from a young age so that it becomes easier to understand complicated concepts correlated with it in the future.

We use this concept in various ways, for example, while playing sports or driving a car, even without the knowledge of actually using it. This article will discuss the Physics equations of motion in detail while having a good look at other similar concepts. These are commonly referred to as the three equation of motion in school-level physics. 

What Are Equations of Motion in Physics?

In kinematics, equations of motion are defined as the motion formula of an object, including velocity, position, and acceleration, which are performed at varying intervals of time. These three motion equations govern an object’s motion in 1, 2, and 3 dimensions.

Simply put, the Physics equations of motion are the set of equations capable of describing a physical system’s behavior in terms of its motion as a function of time. Hence, the equation of motion is the relation between these quantities.

Uniform Acceleration’s Equations of Motion

Uniform acceleration in equations of motion includes three different equations. These are called the laws of constant acceleration. Therefore, we can utilize these equations for deriving components such as velocity (initial and final), acceleration (a), displacement (s), and time (t). 

Hence, these are applied only where there is a constant acceleration and the motion is in a straight line. The following are known as the 3 equations of motion used for uniformly accelerated motion:

• The first equation of motion: v=u+at
• Second equation of motion: s=ut+1/2at2
• Third equation of motion: v² = u² + 2as

where,

s = displacement

u = initial velocity

v = final velocity

a = acceleration

t = time of motion

We can call these quantities SUVAT. It stands for displacement, initial velocity, final velocity, acceleration, and time of motion.

The Three Equations of Motion

When an object moves with uniform acceleration in a straight line, its motion can be fully described using the three equations of motion. These are also called the 3 equations of motion or the three equation of motion.

Each equation connects different motion quantities such as velocity, acceleration, displacement, and time. These equations are used together as a motion formula set in numerical problems.

Equation of Motion	Formula	What It Tells Us	When It Is Used
First Equation of Motion	v=u+at	This is the equation for velocity. It shows how the final velocity changes when an object moves with uniform acceleration.	Initial velocity (u) is knownAcceleration (a) is knownTime (t) is knownFinal velocity (v) needs to be calculated
Second Equation of Motion	s=ut+12at2	This is the equation for displacement. It tells us how far an object travels in a given time with uniform acceleration.	Time (t) is knownInitial velocity (u) is knownAcceleration (a) is knownDistance travelled (s) needs to be calculated
Third Equation of Motion	v2=u2+2as	This equation relates velocity and displacement without using time.	Time is not givenVelocity and distance are involvedUseful when calculating final or initial velocity directly

Here is a section dedicated to the ‘derivation of equation of motion’.

First Equation of Motion – Derivation

The following are the techniques used for deriving equations of motion:

• Derivation of motion equations using the Simple Algebraic Method
• Derivation of motion equations using the Graphical Method
• Derivation of motion equations using the Calculus Method

They all have three equations of motion derivations. Let us see them below:

First Equation of Motion – Derivation

1. Simple Algebraic Method

We all know that the rate of velocity change is based on the acceleration of the body.

Consider a body that contains ‘m’ mass and ‘u’ initial velocity. After some time, ‘t’, it attains its final velocity ‘v’. This final velocity is due to the acceleration ‘a. Hence,

Acceleration = Final Velocity (or) Initial Velocity / Time taken

a = v – u/t or at = v – u

v = u + at 

2. Graphical Method

Take a look at the following graph:

From that,

OD = u; OC = v and OE = DA = t

Uniform acceleration = a

Initial velocity = u

Final velocity = v

Let OE = t (time)

From the graph:

BE = AB + AE 

v = DC + OD (QAB = DC & AE = OD) 

v = DC + u QOD = u 

v = DC + u ……. (1)

Now, 

a = v – u/t 

a = OC – OD/t = DC/t 

at = DC …….. (2)

Therefore, by substituting DC from (2) in (1), we will get:

v = u + at 

3. Calculus Method

We know that the rate of velocity change describes acceleration,

a = dv / dt 

adt = dv 

Integrate both sides:

at = v – u 

Rearrange the above-mentioned equation and we will get:

v = u + at 

Second Equation of Motion – Derivation

1. Simple Algebraic Method

Consider the distance ‘s’.

We all know that,

Velocity = Distance / Time 

So, Distance = Average Velocity × Time 

Also,

Average velocity = u + v / 2

Distance s = u + v2 × t

Also, from v = u + at

s = u + u + at /2 × t 

=2u + at / 2 × t 

=2ut + at² / 2 

=2ut / 2 + at² /2 

Hence, s = ut + 1 / 2at²

2. Graphical Method

Let us see the graph given below:

OD = u, OC = v and OE = DA = t

Initial velocity = u

Final velocity = v

Uniform acceleration = a

From the graph, we can see that the distance covered in the given time ‘t’ is equal to the area of a trapezium ABDOE.

In the given time ‘t’, the distance covered is ‘s’.

Distance, s = Area of ∆ ABD + Area of ADOE.

s = 12 × AB × AD + (OD × OE) 

= 12 × DC × AD + u × t [Since AB = DC]

= 12 × at × t + ut [Since DC = at]

= 12 × at × t + ut 

Therefore, we get:

s = ut + 1/2 at²

3. Calculus Method

We know that the rate of change of displacement gives velocity.

We can equate that as:

v = ds / dt 

ds = vdt 

While integrating the equations together,

ds = (u + at) dt 

ds = (u + at) dt = (udt + atdt) 

Integrating both sides:

Simplifying the equation further to get,

s = ut + 1/2 at²

Third Equation of Motion – Derivation

1. Simple Algebraic Method

We have v = u + at; hence, it can be written as v – u / a

Furthermore, we know that, 

Distance = Average Velocity × Time 

Hence, we can write constant velocity as:

Average Velocity = Final Velocity + Initial Velocity / 2 = v + u / 2

Therefore, distance s = v + u / 2 × v – u / a

or s = v2 – u² / 2a

or, we can write that as:

v²  = u² + 2²/2as 

2. Graphical Method

Consider the graph given below:

From the graph,

OD = u, OC = v and OE = DA = t

Initial velocity = u

Final velocity = v

Uniform acceleration = a

From the graph, we can see that the distance covered in the given time ‘t’, forms the area of a trapezium ABDOE.

In the given time ‘t’, the distance covered is ‘s’.

Therefore, Area of trapezium ABDOE = 1 / 2 × ( Sum of the Parallel Slide + Distance between the Parallel Slides)

Distance, s = 1 / 2 × DO + BE × OE = 12 u + v × t ……(3)

From the 2nd equation: a = v = ut,

t = v – ua …… (4)

Now, substitute the 4th equation in the 3rd one. We get:

s = 12 u + v × (v – ua),

s = 12 × a × v + u (v – u) 

2as = v + u (v – u) 

2as = v²  – u²

Therefore, we can write it as:

v²  = u² + 2as 

3. Calculus Method

a = dv / dt 

The rate of change of displacement gives velocity:

v = ds / dt 

Multiply the above equations. We get:

a × ds / dt = v × dv / dt 

Integrating the above equation,

as = v²  – u²/ 2 

The final equation will be:

v² = u²+ 2as 

Applications of Equations of Motion

Physics equations of motion have numerous real-world applications. The following are some of them:

• Equations of motion make it easier to predict the time of the body’s motion just by reckoning its final and initial velocities with its acceleration. It is not necessary to have knowledge of the body’s travel time.
• It makes it easier for us to predict the distance covered by the body while providing initial and final velocities with acceleration. It is not necessary to have knowledge of the time taken by the body in travel.
• Motion equations will greatly help us obtain the final velocity value of the body. It requires acceleration and initial velocity values.

Conclusion

All in all,  the equations of motion in physics Physics equations of motion are considered the relations that help us relate the physical quantities, namely distance, time taken, acceleration, and initial and final velocity. These equations are capable of facilitating the calculations regarding uniformly accelerated motion along the straight line.

In this blog, we have understood all the concepts and derivations essential for this topic.

Frequently Asked Questions