Question

Define the following terms as used in motion
1. Acceleration 2.uniform acceleration 3.retardation 4. Uniform retardation. 5 instantaneous speed. 6.Instantaneous velocity
2. Derive other 2 equations of motion using the two equations derived above.

Answer 1

Definitions :-

❶

$\tt{\longrightarrow{1. \; Acceleration }}$

The rate of change of velocity per unit of time is called as Acceleration .

♦ The S.I. Unit of Acceleration is m/s^2 .

♦ This definition is represented as ;

$\boxed{\large{\tt{ Acceleration \: = \dfrac{ \Delta V}{ \Delta T}}}}$

Here, ∆ is called as delta .

" ∆ " represents change

$\tt{\longrightarrow{2. \; Uniform \;Acceleration }}$

Equal change in velocity in equal intervals of time is known as uniform acceleration .

$\tt{\longrightarrow{3. \; Retardation }}$

Negative acceleration is known as Retardation .

♦ Acceleration can be positive or negative .

♦ Example : when we will apply brakes in a car the acceleration which is caused is known as retardation .

$\tt{\longrightarrow{4.\; Uniform \; retardation}}$

Negative uniform acceleration is known as Uniform retardation .

$\tt{\longrightarrow{5.\; Instantaneous \; Speed}}$

The speed of the body at any instant is known as Instantaneous speed .

$\tt{\longrightarrow{6. \; Instantaneous \; Velocity}}$

The velocity of the body at any instant is known as Instantaneous Velocity .

❷

Derivation of the 3 equations of motion ;

The 3 equations of motion are :-

$\tt{1.\: v = u + at }$

$\tt{2. \: s = ut + \dfrac{1}{2} a {t}^{2}}$

$\tt{3.\: {v}^{2} - {u}^{2} = 2as}$

Here, is the derivation

We know that ;

Acceleration is the change of velocity per unit of time .

This is represented as ,

$\tt{ a = \dfrac{ \Delta v}{ \Delta t}}$

Change in velocity = initial velocity - Final velocity

∆v = u - v

$\tt{ a = \dfrac{ v - u}{t}}$

$\tt{ at = v - u }$

$\tt{ u + at = v }$

$\tt{\implies{ v = u + at}}$

$\red{ 1st \; equation \; of \; motion \; is \; derived }$

Similarly,

According to physics ;

$\tt{ Average \: velocity = \dfrac{ v - u}{t}}$

According to Maths ;

$\tt{ Average \: velocity = \dfrac{ v + u}{2}}$

consider this as statement - 1

we know that ;

Distance = Average velocity x time

From 1st equation of motion :-

$\tt{ v = u + at }$

$\tt{ t = \dfrac{v - u}{a}}$

consider this as statement - 2

Now substitute the values in the formula from the above 2 statements

s = v x t

$\tt{ s = ( \dfrac{ v + u}{2}) \times ( \dfrac{ v - u}{a})}$

$\tt{ s = \dfrac{ {v}^{2} - {u}^{2}}{2a}}$

$\tt{ 2as = {v}^{2} - {u}^{2}}$

$\tt{\implies{ {v}^{2} - {u}^{2} = 2as}}$

$\red{ 3rd \; equation \; of \; motion \; is \; derived }$

Similarly,

Distance = Average velocity x time

s = v x t

As we know that ;

$\tt{ s = \dfrac{ u + v}{2} \times t}$

From, v = u + at

$\tt{ s = \dfrac{ u + u + at }{2} \times t }$

$\tt{ s = \dfrac{ 2u + at }{2} \times t }$

$\tt{ s = \dfrac{ 2ut + a{t}^{2}}{2}}$

$\tt{ s = \dfrac{ 2ut}{2} + \dfrac{ a {t}^{2}}{2}}$

$\tt{ s = ut + \dfrac{1}{2} a{t}^{2}}$

$\red{ 2nd \; equation \; of \; motion \; is \; derived }$

Hence derived all equations