# Newton’s Equations of Motion     ## Introduction to Newton’s Equations of Motion

In Newtonian mechanics, equations of motion describe the behavior of a physical system in terms of its motion as a function of time. More particularly, the equations of motion describe the attribute of a physical system as a set of mathematical functions in terms of dynamic variables. However, these variables are spatial coordinates and time that may include momentum components as well.

Equations of motions are mathematical formulas that describe the position, velocity, or acceleration of a body relative to a given frame of reference.

If the position of an object changes with respect to a reference point then it is said to be in motion w.r.t. that reference while if it does not change then it is at rest w.r.t. that reference point. For a better understanding or to deal with the different situations of rest and motion, we derive some standard equations relating terms distance, displacement, speed, velocity, and acceleration of the body by the equation called equations of motion.

### Three Equations of Motion

In the case of motion with uniform or constant acceleration (one with equal change in velocity in equal interval of time), we derive three standard equations of motion which are also known as the laws of constant acceleration. These equations contain quantities displacement(s), velocity (initial and final), time(t), and acceleration(a) that govern the motion of a particle. These equations can only be applied when the acceleration of a body is constant and motion is a straight line. The three equations are,

v = u + at

v2 = u2 + 2as

$S=ut+\frac{1}{2}at^2$

Where,

s = The total displacement

u = Initial velocity

v = Final velocity

a = Acceleration

t = Time of motion

### Derivation of Equation of Motions

Now let's start the derivation with one of the simple equations of motion i.e., v=u+at where u is the initial velocity, v is the final velocity and a is the constant acceleration.

Assuming that a body started with initial velocity “u” and after time t it acquires final velocity v due to uniform acceleration a.

We know that the acceleration of a moving particle is defined as the rate of change of velocity, also which is given by the slope of the velocity-time graph.

Therefore, according to the definition of acceleration and the v-t graph we get,

$\Rightarrow \text {Acceleration}=\frac{\text{Change in velocity}}{\text{Total time taken}}$

$\Rightarrow \text {a}=\frac{\text{(Final velocity)-(Initial velocity)}}{t}$

$\Rightarrow \text {a}=\frac{\text{(v)-(u)}}{t}$

On simplifying the above equation we get:

=v=u+at

Now to derive the second equation again suppose a body is moving with initial velocity u after time t its velocity becomes v. The displacement covered by them during this interval of time is S and the acceleration of the body is represented by a.

Explanation: We know the area under the velocity-time graph gives the total displacement of the body. Thus area under the velocity-time graph is the area of trapezium OABC.

Also area of trapezium = $\frac{1}{2}(\text{sum of parallel sides})\times(\text{Height})$

Sum of parallel sides = OA + BC = u + v and here, height = time interval t

Thus,area of trapezium =  $\frac{1}{2}(\text{u})\times(\text{v})t$

Substituting v=u+at from the first equation of motion we get,

Displacement =S =area of trapezium = $\frac {1}{2}$ (u + {u + at}) t

S= $\frac {1}{2}$ (2u + at)t = ut + $\frac {1}{2}$ at2

This is called the second equation of motion and is the relation between displacement S, initial velocity u, time interval t, and acceleration ‘a’ of the particle.

Now in order to derive the third equation again use

Displacement = S =area of trapezium = $\frac {1}{2}$ (u + v) x t

From first equation v=u+at we get v - u= at $\frac {v-u}{a} = t$

Substituting the value of t in S = $\frac {1}{2}$ (u+v) xt

We  get S = $\frac {1}{2}$ (u+v){$\frac {v-u}{a}$}= $\frac {v^2 - u^2}{2a}$

⇒ 2as = v2- u2

⇒ v2 = u2 + 2as

Which is the third equation of motion and is the relation between final velocity v, initial velocity u, constant acceleration a, and displacement S of the particle.

We can now also calculate the displacement of particles during the nth second, using these equations of motion derived above.

Displacement for the nth second can be calculated by using the formula given below:

sn= u + $\frac {1}{2}$  a(2n-1)

This equation is often regarded as a modified form of the second equation of motion

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1. What are the Three Equations of Motion also State How They can be Derived with the Help of Graph?

Three Equations of Motion are v = u + at; s = ut + (1/2) at2 and v2 = u2 + 2as and these can be derived with the help of velocity time graphs using definition acceleration. Here, acceleration is rate of change of velocity thus it is given by slope of v-t graph we use this to derive the first equation of motion while area under v-t graph gives total displacement and this is used to derive second and third equation of motion.

2. What is the Difference Between Speed and Velocity?

Speed is scalar quantity, is the rate at which an object covers its distance and average speed is the ratio of total distance by total time. On the other hand, velocity is a vector quantity and is rate of change of position of an object and average velocity is ratio of total displacement by total time.

3. What is a Kinematic Motion Equation?

The three equations of motion are known as kinematic equations. Quantities associated with these kinematics equations are - displacement (and distance), velocity (and speed), acceleration, and time.

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