**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 Physics equations of motion in detail while having a good look at other similar concepts.

**Physics Equations of Motion – Definition **

In kinematics, equations of motion are defined as the basic motion concept 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 the three equations:

**The first equation of motion:**v = u + at**Second equation of motion:**s = ut + 12 at^{2}**Third equation of motion:**v^{2}= u^{2}+ 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.

**Equations of Motion – Derivation Types**

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**

**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

**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

**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**

**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 + at^{2 }/ 2

=2ut / 2 + at^{2 }/2

Hence, s = ut + 1 / 2at^{2 }

**Graphic 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’ form 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 at^{2 }

**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 at^{2 }

**Third Equation of Motion – Derivation**

**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 – u^{2 }/ 2a

or, we can write that as:

v^{2 } = u^{2 } + 2^{2}/2as

**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 = v^{2 } – u^{2 }

Therefore, we can write it as:

v^{2 } = u^{2 }+ 2as

**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 = v^{2 } – u^{2}/^{ }2

The final equation will be:

v^{2} = u^{2}+ 2as

**Equations of Motion Applications**

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, 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 **

**Q1. Mention the alternate names used for the equations of motion.**

**A:** The first equation of motion, v = u + is referred to as the velocity-time relation. On the other hand, the second equation of motion is s = ut + 1 / 2at^{2} can be called the position-time relation. Likewise, we call the third equation of motion, v^{2} = u^{2}+ 2as, position – velocity relation.

**Q2. Is it possible for equations of motion to be applied to a freely falling body?**

**A:** Yes, it is possible for equations of motion to be applied on a freely falling body, given the acceleration of the body is uniform. This is the acceleration due to gravity. Also, during this free fall, the body travels along a straight vertical line.

**Q3. Give the conditions for applying equations of motion.**

**A:** There are only two conditions that need to be followed. They are:

- The acceleration of the body must be uniform.
- The object should travel along a straight line.

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