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SS2: PHYSICS - 1ST TERM

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  1. Scalars & Vectors | Week 1
    5 Topics
    |
    1 Quiz
  2. Equations of Motion | Week 2
    3 Topics
    |
    1 Quiz
  3. Projectile | Week 3
    5 Topics
  4. Equilibrium of Forces I | Week 4
    4 Topics
  5. Equilibrium of Forces II | Week 5
    4 Topics
  6. Stability of a Body | Week 6
    4 Topics
    |
    1 Quiz
  7. Simple Harmonic Motion (SHM) | Week 7
    4 Topics
  8. Speed, Velocity & Acceleration & Energy of Simple Harmonic Motion | Week 8
    5 Topics
    |
    1 Quiz
  9. Linear Momentum | Week 9
    6 Topics
    |
    1 Quiz
  10. Mechanical Energy & Machines | Week 10
    2 Topics
    |
    1 Quiz



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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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