**Verification of three equations of motion**

**Verifications of Newton’s three equations of motions**

**Three equations of motions** -These **three equations of motions** were discovered by the great British scientist so these **three equations of motions** are also called **Newton’s three equations of motions**. In this post, you will study the verifications of **Newton’s three equations of motions** by the graphical method and algebraic method. Before we study** three equations of motion** lets make a flashback, the relationship between distance and time is that distance is the product of **speed **and time, you already have studied in junior classes.

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When the direction of** moving** object is taken under consideration then the distance is substituted by displacement, **speed** by **velocity** and since time is the scalar quantity it remains the same. Displacement is the product of** velocity** and time. This formula has its limitation if the speed of the object varies with time. If a bus moves in the way with variable **speed** and reached to a destination within a certain time period. Then how will you calculate the total distance traveled? Yes, here we are needed to apply one or a couple of the** three equations of motions** to calculate the total distance. This formula, distance = **speed** × time is applied only when **speed** is constant. In **class 9,** before you study **three equations** **of motions** you have to study now one more physical quantity i.e acceleration.

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**Acceleration.** Acceleration is the rate of change of **velocity**, in simple words, the **velocity** **of speed** is the acceleration or the rate at which the **velocity** varies.

Let u is the initial **velocity** of an object and in time t its velocity become v

Then the change in its **velocity** = v – u

Where ‘a’ is the acceleration of the object.

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**Three equations of motion** can be verified by three methods, here are given two methods, Algraibic and graphical, the third way is by calculus, you will study in class 11.

**Verifications of Newton’s three equations of motions**

**The first equation of motion**.

**Algebraical method of verification of the first equation of motion.**

It is derived from the definition of the acceleration

v = u + at⇒**First equation of the motion**

**Verification of the first equation of motion by graphical method.**

Velocity and time graph of an object is drawn as following in which velocity is shown on y-axis and time is shown on the x-axis

Graphically, the acceleration is shown by the slope of the graph

DE = OA = u, BE = v and AD = OE =t

**The second equation of motion**

**Verification of the second equation of motion by the algebraical method.**

Displacement (s)= **average velocity** ×time taken

**Average velocity** = (v + u)/2 and time = t

**The verification of the second equation of motion by graphical method.**

If the velocity is constant, then the displacement covered is given by DE × OE= area of rectangle AOED

Therefore the total displacement covered by the object in the velocity-time graph can be shown by sum of the area of rectangle and triangle.

Displacement = Area of rectangle AOED + Area of triangle ABD

AD = OE =t, OA = DE = u, BD = BE – DE = v – u

From (i)

v –u = at

**The second equation of motion**

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**Third equation of motion.**

**Verification of the third equation of motion by the algebraic method**

The displacement = average velocity × time period

average velocity = (v + u)/2, time = t

From (i), t = (v – u)/a

v² – u² = 2as

**Verification of the third equation of motion by the graphical method.**

Displacement = Area of trapezium ABEO

AO = u, BE =v, OE = t

From (i) substituting t = (v – u)/a

v² – u² = 2as

**⇒Third equation of motion**

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**Verification,of three equation of Motions by the method of calculus**

First equation of motion: Let an object ‘s initil velocity is u and within time t ,it achieves velocity v

The rate of change of velocity(v) is acceleratiom(a)

dv = a dt

Integrating both sides

∫dv = ∫a.dt

This is the case when acceleration is uniform(constant) and initial velocity is u and final velocity is v

v – u = a[t-0]

v-u = at

**v-u = at,** is the first equation of motion

**Verification of three equation of motion by the method of calculus:**

**Verification of first equation of motion by the method of calculus:**

We know the acceleration = Rate of change of velocity

Let there occures a small change (dv) in velocity within small time span of dt,then acceleration a, can be written as follows

Integrating LHS with respect to time within the limits of time from 0 to t and Integrating RHS with respect to velocity within the limits of velocity from u to v

Since acceleration is constant here

a(t -0) = (v – u)

**at = v – u **

**First equation of motion v -u= at is verified**

**Verification of second equation of motion by the method of calculus:**

We know displacement is the minimum distance travelled by an object in unit time

Let a small distance ds is travelled by an object within a small time span dt

Integrating LHS with respect to displacement within the limits of displacement from 0 to s and Integrating RHS with respect to time within the limits of time from 0 to t

u and a supposed as a constant value

Here it is the verified third equation of motion

**Verification of third equation of motion by the method of calculus:**

For driving third equation,we are needed to define acceleration and velocity

From (i) and (ii) equation, we get

Integrating LHS with respect to displacement within the limits of displacement from 0 to s and Integrating RHS with respect to velocity within the limits of time from u to v

**v² – u² = 2as**

Third equation of motion **v² – u² = 2as **is varified

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