Chapter 2 Motion Along a Straight Line

In this chapter we will study how objects move along a straight line

The following parameters will be defined:

For constant acceleration we will develop the equations that give us the velocity and position at any time..

(1) Displacement (2) Average velocity (3) Average speed

(4) Instantaneous velocity

(5) Average and instantaneous acceleration

Chapter 2 Motion Along a Straight Line

Position vector

• Its magnitude is the distance between the object and the origin.

• Its direction is positive when the object is in the positive

side of axis, and negative when the object is in the negative side.

It consists of:

• the origin (or zero point),

• a coordinate axis: the direction along it is positive. The other direction is negative

Position and Displacement.

Scalars and Vectors

• A scalar quantity can be described by its magnitude only

• A Vector is described with both its magnitude and direction.

A vector can be represented by an arrow:

magnitude

direction

The velocity is a vector.

Displacement and acceleration are also vectors.

E N

W

S

Displacement. If an object moves from position x₁ to position x₂ , the change in position is described by the displacement

2 1

x x x

   ^O

. .

^{x1 x}

.

^{2 x-axis}

motion Δx

• It is the change of the object’s position

• It points from the initial position to the final position of the object

• Its magnitude equals the distance between the two positions.

• SI Unit of Displacement: meter (m)

Note: The actual distance for a trip is irrelevant as far as the displacement is concerned.

Consider as an example the motion of an object from an initial position x₁ = 5 m to x_c = 25 m and then back to x₂ = 12 m. Even though the total distance covered is 33 m the displacement then is Δx = +7 m.

. . .

O x₁ x₂

x-axis Δx

x_c

20 m

13 m

. . .

O x₁ x₂

x-axis Δx

For example if x₁ = 5 m and x₂ = 12 m then Δx = 12 – 5 = +7 m.

The positive sign of Δx indicates that the motion is along the positive x-direction.

If instead the object moves from x₁ = 5 m and x₂ = 1 m then Δx = 1 – 5 = - 4 m.

The negative sign of Δx indicates that the motion is along the negative x-direction.

. . .

O x₂ x₁

x-axis

Δx

Displacement is a vector quantity that has both magnitude and direction. In this restricted one-dimensional motion the direction is described by the algebraic sign of Δx.

Question What is the magnitude of the displacement in this case?

The initial and the final positions of a particle moving along x-axis are X₁ = -73 m, X₂ = 97 m, then its displacement ∆x equals:

Example :

(A) +24 m (B) +170 m (C) -170 m (D) -340 m

Average Velocity

• x₂ and x₁ are the position vectors at the final and initial times

• Angle brackets denotes the average of a quantity.

• SI Unit of Average Velocity: meter per second (m/s)

m/s 0 12

4

0 - 48

4

48m 2x4

4x4 4)

(t x

0 0

2x0 4x0

0) (t

x

1 2

1 2

2 2

2

1 2

1

 

 

 



 





















t t

x x

t v x

t at

t at

avg

Average Speed

speed: the magnitude of velocity

Average speed is always positive

Average velocity could be negative, positive or zero

depending on the direction of the partial velocities

Example

A bike travels12km in 90 mins. Its average speed is:

(A) 8 km/h

(B) 18 km/h

(C) 28 km/h

(D) 48 km/h

km/h 5 30

. 0

15 tan

h 5 . 60 0

min 30 30



 









t

ce v dis

t

avg

m/s

3 )

0 (

2 

3

  

 t v t

v

Instantaneous Velocity and Speed

• It is the time derivative of the object's position.

• It is obtained at any instant from the average velocity by shrinking the time interval t closer and closer to zero

• Instantaneous speed (speed) is the magnitude of the instantaneous velocity vector

The position of an object is given by x = 4t³ - 2t² + 2.5t, where x and t are in SI units. What is the instantaneous velocity of the object when t = 0.25 s.

Example :

(A) 3.75 m/s (B) 5.45 m/s (C) 2.25 m/s (D) 4.55m/s

Example :

The position of a particle moving on an X- axis is given by x= 4+7t-t ² with x in (m) t in (s). The velocity at 3 s is :

(A) 4 m/s

(B) 2 m/s

(C) 1 m/s

(D) 0.4 m/s

Average Acceleration

We define the average acceleration a_avg between t₁ and t₂ as:

2 1

avg

2 1

v v v

a t t t

 

 

 

Units: m/s

²

Example :

2 1

2

1 2

2 2

1 1

m/s 5 4

20 3

8

20 0

8 /

0

3 /

20



 

 

 



 



 









t t

v v

t a v

s t

at s

m v

s t

at s

m v

avg

Instantaneous Acceleration

If we take the limit of a_avg as Δt → 0 we get the instantaneous acceleration a, which describes how fast the velocity is changing at any time t.

Units: m/s

²

Example :

(E)

We will develop the

equations that describe motion with a constant

a

(a special case).

We assume that initially (at time

t

=0 ) the particle is at

x

_o and moves with

v

_o^{, and}

some time

t

later (final state) the particle is at

x

and move with

v

^.

a = constant

Motion with Constant Acceleration

Motion with Constant Acceleration

We will develop the equations that describe motion with a constant a ( a special case ).

We assume that initially (at time t=0 ) the particle is at x0 and moves with v0, and some time t later (final state) the particle is at x and move with v.

The first two equations are originals, but all others are driven from the first two.

(1)

(2)

(3)

(4)

(5)

بلاط نم بولطم سيل ةكرحلا تلاداعمل يضايرلا جاتنتسلاا Phys 101

.

(1)

(2)

(3)

(4)

(5)

?

s

5 . 8

m/s

8 . 2

40m

0 0











v t v

x x

s m

v

v

t v v

x x

a

/ 61

. 6

5 . 8 ) 8 . 2 2 (

40 1

) 4 ( )

2 ( ) 1

(

0

0

0 0













2 0

/ 448

. 0

5 . 8 62

. 6 8

. 2

) 1 ( )

(

s m

a

a at v

v b











m 32 4

2 x2x 4x4 1

) 2 2 (

1

(?)

s 4

m/s 2

m/s

4

2 0

2 0

0

0 2

0





















x x

at t

v x

x

x-x t

a v

Example

2 0

/ 5

10 0

50

) 1 (

s m

a

a at v

v











Example

m x

x

x x

x x

s t

s m h

km v

s m h

km v

75

6 3600)

20 1000 3600

70 1000 2 (

1

? 6

/ /

20

/ /

70

0

0

0 0































Free Fall

Close to the surface of the Earth all objects move

toward the center of the Earth with an acceleration whose magnitude is constant and equal to 9.8 m/s². We use the symbol

g

to indicate the acceleration of an object in free fall.

Free Fall

• its magnitude is g; it is independent of the object's characteristics, such as mass, density, or shape

• g varies slightly with latitude and with elevation; at the sea level g=9.8 m/s² (or 32 ft/s²)

• The equations of motion for constant acceleration also apply to free fall near Earth's surface either up or down

• The directions of motion are now along a vertical y axis: it is +ve for upward motion and –ve for downward motion (a = - g)

Free Fall

a y

gg

Free Fall Equations

Example

 

 

 

m/s 83

. 6

0.84

2 . 39

2 . 39 84

. 0

2 . 39 16

. 0 2

. 39 16

. 0

2 8

. 9 2

0.4

2

(3)

2

0 2

0

2 0

2 0 2

0 2

0 2

0

2 0 2

0

0 2

2 0

0 2

0 2







































v v

v

v v

v v

v v

y y

g v

v

y y

g v

v

m

2

?

m/s

8 . 9

4 . 0

0 0

2 0









y-y v

g v

v

A stone is released from rest from the edge of a building roof 190 m above the ground. Neglecting air resistance, the speed of the stone, just before striking the ground, is

Example

 

s m v

why v

v v

y y

g v

v

/ 61

3721

3724 190

8 . 9 2

190 8

. 9 2

) (

190 8

. 9 2

0

(3)

2

2 2

2

0 2

0 2





































?

m 190

m/s

8 . 9

0

0

2 0









v y-y

g

(A)120 m/s

v

(B)190 m/s (C) 43 m/s (D) 61 m/s