Lesson 2, Topic 1
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# Speed, Velocity & Acceleration (Equations of Motion)

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Speed is defined as the distance moved per time taken. It is a scalar quantity i.e. it has only magnitude no direction.

Speed = $$\frac{distance \: moved}{time \: taken}\\ =\frac{kilometre}{hour}$$

Its unit is Kmh-1 or ms-1

Velocity is the time rate of increase displacement. It is a vector quantity. It is measured in ms-1.

v = $$\frac{increase \: in \: displacement}{time }$$

Acceleration is the time rate of increase velocity. It is a vector quantity and its unit is ms-2.

a = $$\frac{increase \: in \: velocity}{time }$$

a = $$\frac{V \:- \:U}{t}$$

v = Final velocity

u = Initial velocity

t = time in seconds

Retardation is the time rate of decrease in velocity. It is a vector quantity and measured in ms-2.

a = $$\frac{U\: -\: V}{t}$$

### Derivation of Equations of Motion:

Derivation of First Equation of Motion:

Suppose a body is moving with uniform acceleration, a, and increase in velocity, v, in time, t, in seconds.

acceleration of the body, a.

a = $$\frac{change\: in \: velocity}{time }$$

a = $$\frac{v \:- \:u}{t}$$

∴ at = v – u

$$\scriptsize v = u \: + \: at\:……(1)$$

v – Final velocity

u – Initial velocity

a – acceleration

t – time in seconds

This is called the first equation of motion.

### Derivation of Second Equation of Motion:

If the velocity of the body increases as it undergoes uniform acceleration, the average velocity is equal to half of the initial and final velocities.

i.e. average velocity = av = $$\frac{u \:+ \:v}{2}$$

The distance covered, s, during the journey is

s = average velocity x time

s = $$\frac{u \:+ \: v}{2} \scriptsize \times t$$ …………….*

From equation (1), v = u + at; substituting for v in equation *; then

s = $$\left ( \frac{u \:+\: u\: +\: at}{2} \right) \scriptsize\: \times\: t$$

s = $$\left (\frac{2u \: +\: at}{2}\right) \scriptsize\: \times \: t$$

s = $$\left (\scriptsize u \: +\: \normalsize \frac{at}{2} \right) \scriptsize\: \times\: t$$

s = $$\scriptsize ut \:+ \:\normalsize \frac{at^2}{2}\: \scriptsize……(2)$$

This is the second equation of motion.

### Derivation of Third Equation of Motion:

Method 1:

Using the above equations (1) and (2), and eliminating t,

v = u + at

Squaring both sides,

v² = (u + at)²

v² = (u + at)(u + at)

v2= u2 + 2aut + a2t2

v² = u² + 2aut + (at)²

At this point, you will then factorise 2a from 2aut + (at)².

v2 = $$\scriptsize u^2 \: + \: 2a(ut \: +\: \normalsize \frac{1}{2}\scriptsize at^2)$$

From equation (2)

S = $$\scriptsize ut \:+ \:\normalsize \frac{at^2}{2}$$

Let $$\: \scriptsize (ut \:+\: \normalsize \frac{1}{2}\scriptsize at^2 )$$ be replaced by S, then

v2 = u2 + 2aS.

Method 2

From equation * S = $$\frac{u \;+ \;v}{2} \scriptsize \times t$$

From the first equation of motion, we know that

V = u + at

If we make t the subject of the formula, we get

t = $$\frac{v \; -\; u}{a}$$

Substituting the value of t in equation *, we get

S = $$\left( \frac{u \;+ \;v}{2} \right)\left( \frac{v \; -\; u}{a}\right)$$

S = $$\frac{v^2 \;- \;u^2}{2a}$$

Multiply both sides by 2a

2as = v2 – u2

Rearranging we get

$$\scriptsize v^2 = u ^2 \: + \: 2as \:……(3)$$

The four equations of motions are;

(1) v = u + at

(2) s = $$\scriptsize ut \: +\: \normalsize \frac{at^2}{2}$$

(3) $$\scriptsize v^2 = u ^2 \: + \: 2as$$

(4) s = $$\frac{u\: + \:v}{2} \scriptsize\: \times\: t$$

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