PHY 130 - Chapter 2 -Kinematics in One Dimension

1

PHY130

Chapter 2 Kinematics in One Dimension

Assoc. Prof. Dr. Ahmad Taufek Abdul Rahman PhD (Medical Physics), University of Surrey, UK

M.Sc. (Radiation Health Physics), UTM

B.Sc. Hons. (Physics & Math), UTM

[email protected]

[email protected]

https://www.facebook.com/DR.ATAR.UiTM

(HP) 012476764

(O) 064832154/2115

(O) 066632427

ROOM-022 / level 4 (K.Pilah)

2

2.0 KINEMATICS IN ONE DIMENSION

2.1 Scalars and vectors

2.2 Linear motion parameters

2.2.1 Definition of linear motion

parameters

2.2.2 Average and instantaneous

velocity

2.2.3 Average and instantaneous

acceleration

2.3 Graph of linear motion

2.3.1 Displacement time graph

2.3.2 Velocity time graph

2.4 Linear motion with constant

acceleration

2.5 Free fall motion

Chapter 2

3

Chapter 2

Scalars and Vectors Scalar quantity is defined as a quantity with magnitude only.

e.g. mass, time, temperature, pressure, electric current, work, energy and etc.

Mathematics operational : ordinary algebra

Vector quantity is defined as a quantity with both magnitude & direction.

e.g. displacement, velocity, acceleration, force, momentum, electric field, magnetic field and etc.

Mathematics operational : vector algebra

4

Chapter 2

Vectors

Vector A Length of an arrow magnitude of vector A

Direction of arrow direction of vector A

displacement velocity acceleration

s

v

a

s av

s (bold) v (bold) a (bold)

vv

aa

5

Chapter 2

Vectors

Two vectors equal if both magnitude and direction are the same, as shown below.

P

Q

QP

6

Chapter 2

Vectors If vector A is multiplied by a scalar quantity k

Then, vector A is

if k = +ve, the vector is in the same direction as vector A.

if k = - ve, the vector is in the opposite direction of vector A.

Ak

Ak

A

A

7

Chapter 2

Direction of Vectors

Can be represented by using:

a) Direction of compass, i.e east, west, north, south, north-east,

north-west, south-east and south-west

b) Angle with a reference line

e.g. A man throws a stone with a velocity of 10 m s-1, 30 above

horizontal.

30

v

x

y

0

8

Chapter 2


Can be represented by using:

a) Direction of compass, i.e east, west, north, south, north-east,

north-west, south-east and south-west

b) Angle with a reference line

e.g. A man throws a stone with a velocity of 10 m s-1, 30 above

horizontal.

30

v

x

y

0

9

Chapter 2


c) Cartesian coordinates

2-Dimension (2-D)

m) 4 m, 2(),( yxs

10

Chapter 2


c) Cartesian coordinates

3-Dimension (3-D)

m 2) 3, 4,(),,( zyxs

11

Chapter 2


d) Polar coordinates

12

Chapter 2


e) Denotes with + or signs.

+

+ -

-

13

POSITION, DISTANCE AND DISPLACEMENT

Coordinate system defines position

Distance length of actual path between two points

(SI unit = meter, m)

Scalar quantity

Displacement distance between initial point and final point in a straight line (change in position)

Change in position = (final position initial position)

x = xf xi (SI unit = meter, m)

Vector quantity

Chapter 2

14


Example 1:

The purple doted line is a distance, and the green doted line shows a

displacement.

Chapter 2

15


Chapter 2

Before describing motion, you

must set up a coordinate system

define an origin and a positive direction.

The distance is the total

length of travel; if you

drive from your house to

the grocery store and

back, what is the total

distance you traveled?

Displacement is the change in position. If you drive from your house to

the grocery store and then to your friends house, what is your total distance? What is your displacement?

16


Example 2:

An object P moves 20 m to the east after that 10 m to the south and

finally moves 30 m to west. Determine the displacement of P relative

to the original position.

Chapter 2

17

AVERAGE SPEED AND VELOCITY

Average speed distance traveled divided by the total elapsed time (the rate of change of distance)

SI units, meters per second (ms1)

Scalar quantity

Always positive

Chapter 2

timeelapsed

distance speed Average

18


What is the average speed of the red car?

a) 40 mi/h

b) More than 40 mi/h

c) Less than 40 mi/h

Chapter 2

19

AVERAGE SPEED AND VELOCITY Average velocity displacement divided by the total elapsed time

(the rate of change of displacement)

SI units, meters per second (ms1)

Vector quantity

Can be positive or negative

Chapter 2

if

if

avtt

xx

t

xv

timeelapsed

ntdisplaceme velocity Average

20


Chapter 2

Whats your average velocity if you return to your starting point?

What if the runner sprints 50 m in 8 s?

What if he walks back to the starting

line in 40 s?

Can you calculate:

a) What is his average sprint velocity?

b) His average walking velocity?

c) His average velocity for the entire trip?

21

INSTANTANEOUS VELOCITY Instantaneous velocity

This means that we evaluate the average velocity over a shorter and shorter period of time; as that time becomes infinitesimally

small, we have the instantaneous velocity.

Magnitude of the instantaneous velocity is known as the instantaneous speed

Chapter 2

22

INSTANTANEOUS VELOCITY

Chapter 2

23

ACCELERATION Average acceleration the change in velocity divided by the time

it took to change the velocity

SI units meters/(second second), m/s2

Vector quantity

Can be positive or negative

Accelerations give rise to force

Chapter 2

24

INSTANTANEOUS ACCELERATION Instantaneous acceleration - This means that we evaluate the

average acceleration over a shorter and shorter period of time; as

that time becomes infinitesimally small, we have the instantaneous

acceleration.

When acceleration is constant, the instantaneous and average accelerations are equal

Chapter 2

25

ACCELERATION

Acceleration (increasing speed) and deceleration (decreasing speed) should not be confused with the directions of velocity and acceleration:

In 1-D velocities & accelerations can be + or - depending on whether they point in the + or - direction of the coordinate system

Leads to two conclusion

When the velocity & acceleration have the same sign the speed of the object increases (in this case the velocity & acceleration point in the same direction)

When the velocity & acceleration have opposite signs, the speed of the object decreases (in this case the velocity & acceleration point in opposite directions

Chapter 2

26

ACCELERATION

Under which scenarios does the cars speed increase? Decrease?

Chapter 2

27

GRAPHICAL METHODS Displacement against time graph (s-t)

Chapter 2

s

t 0

(a) Uniform velocity

Gradient = constant

28


Chapter 2

s

t 0

(b) The velocity increases with time

Gradient increases

with time

29


Chapter 2

s

t 0

Q

R P The direction of

velocity is changing.

Gradient at point R is negative.

Gradient at point Q is zero.

The velocity is zero.

30

GRAPHICAL METHODS Velocity versus time graph (v-t)

Chapter 2

t1 t2

v

t 0 (a)

t2 t1

v

t 0 (b)

t1 t2

v

t 0 (c)

Uniform

velocity

Uniform

acceleration

B C

A

Explain at A, B and C

Area under the v-t graph = Displacement

31

Example 3:

A toy train moves slowly along a straight track according to the

displacement, s against time, t graph in figure.

a. Explain qualitatively the motion of the toy train.

b. Sketch a velocity (cm s-1) against time (s) graph.

c. Determine the average velocity for the whole journey.

d. Calculate the instantaneous velocity at t = 12 s.

Chapter 2

32

Example 4:

A velocity-time (v-t) graph in figure 3.2 shows the motion of a lift.

a) Describe qualitatively the motion of the lift.

b) Sketch a graph of acceleration (m s-1) against time (s).

c) Determine the total distance travelled by the lift and its

displacement.

d) Calculate the average acceleration between 20 s to 40 s.

Chapter 2

33

Example 5:

An ETS train from Ipoh to Kuala Lumpur running at 30.0 ms1 slows

down uniformly to a stop within 44.0 s. Calculate:

i. the acceleration of the train,

ii. the stopping distance.

Chapter 2

34

MOTION AT CONSTANT ACCELERATION From the definition of average acceleration, uniform (constant) acceleration is given by

where v : final velocity

u : initial velocity

a : uniform (constant) acceleration

t : time

Chapter 2

atuv (1)

t

uva

35

MOTION AT CONSTANT ACCELERATION From equation (1), the velocity-time graph is shown in figure:

From the graph,

The displacement after time, s = shaded area under the graph

= the area of trapezium

Hence,

Chapter 2

tvu2

1s (2)

36

MOTION AT CONSTANT ACCELERATION By substituting eq. (1) into eq. (2) thus:-

From eq. (1),

From eq. (2),

Chapter 2

tatuus 2

1(3)

2

2

1atuts

atuv

t

suv

2

att

suvuv

2

asuv 222 (4)

37

MOTION AT CONSTANT ACCELERATION Notes:

equations (1) (4) can be used if the motion in a straight line with constant acceleration.

For a body moving at constant velocity, ( a = 0) the equations

(1) and (4) become

Therefore the equations (2) and (3) can be written as

Chapter 2

vts constant velocity

atuv 2

2

1atuts asuv 222

uv

38

MOTION AT CONSTANT ACCELERATION

Example 6:

A plane on a runway takes 16.2 s over a distance of 1200 m to take

off from rest. Assuming constant acceleration during take off, calculate

a. the speed on leaving the ground,

b. the acceleration during take off.

Chapter 2

39


Example 7:

A bus travelling steadily at 30 m s1 along a straight road passes a

stationary car which, 5 s later, begins to move with a uniform

acceleration of 2 m s2 in the same direction as the bus. Determine

a. the time taken for the car to acquire the same velocity as the

bus,

b. the distance travelled by the car when it is level with the bus

Chapter 2

40


Example 8:

A particle moves along horizontal line according to the equation

Where s is displacement in meters and t is time in seconds.

At time, t =2.00 s, determine

a. the displacement of the particle,

b. Its velocity, and

c. Its acceleration.

Chapter 2

ttts 23 243

41

FALLING OBJECTS

Free fall is the motion of an object subject only to the influence of

gravity. The acceleration due to gravity is a constant, g.

Chapter 2

42

FALLING OBJECTS

is defined as the vertical motion of a body at constant

acceleration, g under gravitational field without air resistance.

In the earths gravitational field, the constant acceleration

known as acceleration due to gravity or free-fall acceleration or gravitational acceleration.

the value is g = 9.81 m s2

the direction is towards the centre of the earth (downward).

Note:

In solving any problem involves freely falling bodies or free fall motion, the assumption made is ignore the air resistance.

Chapter 2

43

FALLING OBJECTS

Equations of linear motion and freely falling bodies:

Chapter 2

Linear motion Freely falling bodies

atuv gtuv

as2uv 22 gsuv 222

2at2

1uts 2

2

1gtuts

44

FALLING OBJECTS

Assuming air resistance is negligible,

the acceleration of the ball, a = g when the ball moves upward and its velocity

decreases to zero when the ball

reaches the maximum height, H.

Chapter 2

45

FALLING OBJECTS

Example 9:

A ball is thrown from the top of a building is given an initial velocity of

10.0 m s1 straight upward. The building is 30.0 m high and the ball

just misses the edge of the roof on its way down, as shown in figure

3.7. Calculate

a. the maximum height of the stone from point A.

b. the time taken from point A to C.

c. the time taken from point A to D.

d. the velocity of the stone when it reaches point D.

(Given g = 9.81 m s2)

Chapter 2

46

FALLING OBJECTS

Example 10:

A book is dropped 150 m from the ground. Determine

a. the time taken for the book reaches the ground.

b. the velocity of the book when it reaches the ground.

(given g = 9.81 m s-2)

Chapter 2

47

Thank You

Peace cannot be kept by force; it can only be

achieved by understanding.

(Albert Einstein)

PHY 130 - Chapter 2 -Kinematics in One Dimension

Documents

Transcript of PHY 130 - Chapter 2 -Kinematics in One Dimension