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.

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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