Pg. 98

Define Motion

Motions are inferred through direct and indirect evidences

• Watching an object move
• Standing in the bus, watching the trees move backwards, but the co passengers are sitting, the bus boarders move with the bus for the bystander. Refer an object using a reference point, usually origin.

Pg. 99

Motion along a straight line:

Between A , B, C and O

Distance vs Displacement

• Distance is not always equal to Displacement.
• Distance is the total length covered
• Displacement is the shortest distance between initial and final position of the object.
• Displacement can be zero, but not distance.

Pg. 100

Uniform and Non - Uniform motion

• An object covers equal distance in equal intervals of time = Uniform motion
• Unequal distance in equal intervals of time = Non - uniform motion man jogging in a park

Pg 101

Speed

Measuring the rate of motion of an object

• Find the distance travelled by an object in unit time
• v = s / t
• SI unit = metre per second - m / s
• Other units = km / hr, cm / s
• we only need to specify the magnitude

Average speed

• for objects in non - uniform motion
• dividing the total distance travelled by total time taken
• Average Speed= total distance / time taken

Example 8.1

Speed with Direction = Velocity

• It is the speed of an object moving in a definite direction.
• It can be uniform or variable.

Average velocity

• When an object is moving along a straight line at a variable speed, then we can express the magnitude of its rate of motion in terms of average velocity.
• Avg. velocity = arithmetic mean of initial velocity and final velocity for a given period of time.
• v(avg) = (u + v) / 2
• SI unit is same as speed, m/s

Example 8.2

Example 8.3

Pg. 102 - 103

Rate of change of velocity = Acceleration

• During uniform motion, velocity remains constant. Which means there is no change in velocity at any particular time.

• But in non - uniform motion, velocity varies with time.

• Measure of change in the velocity of an object per unit time is known as acceleration.

• Acceleration = change in velocity / time taken

• a = (v-u) / t

• SI unit is m / s^2

• The kind of motion is known as accelerated motion.

Positive and Negative acceleration

• (+) - when acceleration is in the direction of velocity
• (-) - when acceleration is in the opposite direction of velocity

Uniform and Non - uniform acceleration

• When a objects travels in straight line, its velocity increases or decreases by equal amounts in equal intervals of time = Uniform acceleration.

• Ex: A free falling body

• If the velocity changes at a non - uniform rate in equal intervals of time = Non - uniform acceleration.

• Ex: A car travelling in different velocity

Example 8.4

Pg. 104 - 107

Graphical representation of motion

A straight line graph helps in solving the linear equation in 2 variables.

Distance - Time graphs

• Time = x axis
• Distance = y axis

Can be plotted for :

• uniform speed / uniform velocity - a straight line graph
• non uniform speed
• accelerated motion

To determine the speed at a particular point:

• v = (s2 - s1) / (t2 - t1)
• s1, s2 = distance at time t1, t2

Velocity - Time graphs

• Time = x axis
• Velocity = y axis
• [!] For non uniform accelerated motion, the graph can take any form.

To find the displacement for constant speed or velocity

• The product of velocity and time taken gives the displacement of the object moving with uniform velocity.
• AB = points along the line parallel to x axis
• CD = points on the x axis with respect to AB respectively
• s = height of velocity (y axis) * time taken (x axis)
• s = AC * CD
• s = AC * (t2 - t1) – eq.1
• s = area of the rectangle ABDC (shaded)
• s = length * breadth – eq.2
• eq.1 = eq.2

To find the displacement for uniform accelerated motion

• The area under the graph gives the magnitude of the displacement
• A and E = points on the line
• B and C = points on the x axis
• D = between C and E - straight to A
• s = area of rectangle ABCD + area of triangle ADE
• s = (AB * AC) + (1/2 * AD * AE)

Pg. 107 - 110

Equations of motion by graphical method

Equations of motion

uniform acceleration, velocity, acceleration during motion, distance covered, time taken can related as follows

1. v = u + at - velocity time relation
2. s = ut + 1/2 at^2 - position - time relation
3. 2as = v^2 - u^2 - position - velocity relation

Equation of velocity - time relation

Derivation

• Draw AD parallel to OC
• BC = BD + DC = BD + OA
• let BC = v (final velocity) and OA = u (initial velocity)
• v = BD + OA
• v = BD + u
• BD = v - u -----> eq.1
• Acceleration (a) = change in velocity / time taken
• a = BC - CD / AD = BD / AD = BD / OC
• OC = t
• a = BD / t
• BD = at ----> eq.2
• eq. 1 = eq. 2
• v - u = at
• v = u + at

Equation for position - time relation

Derivation

• s - distance travelled
• t - time taken
• s = area of OABC (trapezium)
• s = area of rectangle OADC + area of triangle ABD
• s = OA * OC + 1/2 (AD * BD)
• OA = u. OC = AD = t, BD = at
• s = u * t + 1/2 (t * at)
• s = ut + 1/2 (at^2)

Equation for position - velocity relation

Derivation

• s = distance travelled
• t = time taken
• a = uniform acceleration
• s = area of trapezium OABC = 1/2 × (sum of its parallel sides) × (its height)
• s = ( OA + BC ) * OC / 2
• OA = u, BC = v, OC =t
• s = (u + v) t / 2
• t = v - u / a
• s = (u+v) (u-v) / 2a
• 2as = v^2 - u^2

Example 8.5

Example 8.6

Example 8.7

Pg 110

Uniform circular motion

• [>] When an object moves in a circular path with uniform speed, its motion is called uniform circular motion
• The change in velocity could be due to change in its magnitude or direction or both.
• Example for the magnitude does not change and only the direction of the objects changes:
  • athlete running in a rectangular or hexagonal or octagonal or circular tracks
  • he have to change his direction for 4 or 6 or 8 times and indefinitely in the last one
  • as you increase the number of sides, the number of turns also increases, and reaches infinite in circle.

Example

• discuss throw, throwing a stone tied to a thread after rotating it above our head

• in both cases, when it left to fly, it covers a linear path of acceleration tangential to thee circular path.

• this is because, once it is released, it continues to move in the direction it has been moving at that instant.

• This shows that the direction changes at every instant while moving in a circular path.

• The motion of the athlete along a circular path is known as accelerated motion

Derivation

• circumference of the circle = 2r
• radius - r
• time taken - t
• v = distance / time = 2r / t

Extra

SI Units:

• international decimal system of weights and measures derived from and extending the metric system of units.
• Adopted by the 11th General Conference on Weights and Measures (CGPM) in 1960, it is abbreviated SI in all languages.
• The early international system devised to rectify this situation was called the metre-kilogram-second (MKS) system.
• The CGPM added three new units (among others) in 1948.
• Base units: metre - length, second - time, kilogram - mass, ampere - electric current, kelvin - thermodynamic pressure, candela - luminous intensity, mole - amount of substance. Britannica

Speedometer vs. Odometer

An odometer is like your car's personal travel journal. It keeps a running tally of all the kilometres your vehicle has driven in its lifetime. Whether you're just running errands around town or going on a long road trip, the odometer faithfully records every single kilometre.

The speedometer is like its radar gun. While the odometer tracks distance, the speedometer is all about monitoring your car's current speed at any given moment. They display your speed in either kilometres per hour (km/h) or miles per hour (mph), depending on where you live.

[Reliance General](Discovering the Difference Between Odometer and Speedometer)

Tests and Reference

[Physics Catalyst](Class 9 motion notes Chapter 8 of science NCERT book)