Chapter 3

Position, Displacement, and Average Velocity

For this chapter, we restrict motion in three ways:
1. We consider motion along a straight line only
2. We discuss only the motion itself, not the forces that cause it
3. We consider the moving object to be a particle

A particle is either:

A point-like object (such as an electron)

Or an object that moves such that each part travels in the same direction at the same rate (no rotation or stretching)

Position, Displacement, and Average Velocity

Position is measured relative to a reference point:
 o
The origin , or zero point, of an axis
Position has a sign: o o
Positive direction is in the direction of increasing numbers
Negative direction is opposite the positive
Figure 2-1

Position, Displacement, and Average Velocity

A change in position is called displacement o
∆ x is the change in x , ( final position ) – ( initial position )
Displacement: is the displacement vector which represent the change in the position vector.
∆𝒓 = 𝒓
𝟐
− 𝒓
𝟏
Or can written
Examples:
∆𝒙 = 𝒙
𝟐
− 𝒙
𝟏
A particle moves . . .
o o o
From x = 5 m to x
= 12 m: ∆ x = 7 m (positive direction)
From x = 5 m to x = 1 m: ∆ x = -4 m (negative direction)
From x = 5 m to x = 200 m to x = 5 m: ∆ x = 0 m
Eq. (2-1)

The actual distance covered is irrelevant

Position, Displacement, and Average Velocity
The difference between Distance and Displacement:
Distance is a scalar quantity representing the interval between two points.
Displacement is a vector quantity defined as distance between the initial point and final point of an object.
It must be the shortest interval connecting the initial and final points, that is a straight line .

Position, Displacement, and Average Velocity

Average velocity is the ratio of: o o
A displacement, ∆ x
To the time interval in which the displacement occurred, ∆t
Eq. (2-2)

Average velocity has units of ( distance ) / ( time ) o
Meters per second, m/s

Position, Displacement, and Average Velocity

Average speed is the ratio of: o o
The total distance covered
To the time interval in which the distance was covered, ∆t
Eq. (2-3)

Average speed is always positive (no direction)
Example: seconds.
A particle moves from x = 3 m to x = -3 m in 2 o
Average velocity = -3 m/s; average speed = 3 m/s

Instantaneous Velocity and Speed

Instantaneous velocity , or just velocity , v , is: o o o o
At a single moment in time
Obtained from average velocity by shrinking ∆t
The slope of the position-time curve for a particle at an instant (the derivative of position)
A vector quantity with units ( distance ) / ( time )
• The sign of the velocity represents its direction
Eq. (2-4)

Acceleration


A change in a particle's velocity is acceleration
Average acceleration over a time interval
∆t is
Eq. (2-7)

Instantaneous acceleration (or just acceleration ), a , for a single moment in time is: o
Slope of velocity vs. time graph
Eq. (2-8)

Acceleration

Combining Eqs. 2-8 and 2-4 :
Eq. (2-9)

Acceleration is a vector quantity: o o
Positive sign means in the positive coordinate direction
Negative sign means the opposite o
Units of ( distance ) / ( time squared )

Position, Displacement, and Average Velocity
Example:
Ali walks 8 m north then 8 m south in a total 4 s .
• he travel a distance of 16 m
• his displacement 0 m
• his average speed 4 m
• his average velocity 0 m
© 2014 John Wiley & Sons, Inc. All rights reserved.

Position, Displacement, and Average Velocity
Example :
Calculate the acceleration of a car that travels from 4 m to 16 m in 2 s ?
= ½(16 – 4 )
= 6 m/s 2
© 2014 John Wiley & Sons, Inc. All rights reserved.

Position, Displacement, and Average Velocity

Position, Displacement, and Average Velocity
HW#:
A particle moves along x-axis with a speed given v(t) = 2t 3 + 6t + 6 , find a . The initial speed ?
(ans= 6 m/s) b . The acceleration at t = 2 s of the particle ?
(ans= 30 m/s 2 )
© 2014 John Wiley & Sons, Inc. All rights reserved.

MOTION WITH CONSTANT acceleration(x-axis)
•
Constant acceleration occurs when an object's velocity changes by an equal amount in every equal time period
•
The equations that represent motion of any object at x-axis with constant acceleration :
© 2014 John Wiley & Sons, Inc. All rights reserved.

MOTION WITH CONSTANT acceleration
Example:
A bicycle start its motion from the rest , If its speed after 3 s is 9 m/s , find a . Its acceleration b . The distance it travelled in 4 s
•
Solve a . v f
= v i
+ at ͢ 9 = 0 + a (3) ͢ a = 3m/s 2 b . x f
= x i
+ v i t+ ½ at 2 x f
= 0 + 0 +1/2 (3)(4) 2 x f
= 24 m
© 2014 John Wiley & Sons, Inc. All rights reserved.

MOTION WITH CONSTANT acceleration(x-axis)
Example :
A car travels 22 m in 5 s to increase its speed to 12 m/s .
a . What is the initial speed of the car?
b . Find its acceleration?
© 2014 John Wiley & Sons, Inc. All rights reserved.

MOTION WITH CONSTANT acceleration(x-axis)
Solve :
1. x f
= x i
+ ½ (v f
+ v i
)t
22 = 0 + ½ (12 + v i
)(2)
V i
= 10 m/s
---
2 . v f
= v i
+ at
12=10 + a (5) a = 0.4 m/s 2
© 2014 John Wiley & Sons, Inc. All rights reserved.

MOTION WITH CONSTANT acceleration(x-axis)
HW #:
An automobile moves with a constant speed of 15 m/s
If the driver applies the breaks till it comes to completely stop in 3 s . a . Find the distance that the car moves since the brake?
(ans= 22.5 m)
© 2014 John Wiley & Sons, Inc. All rights reserved.

Free-Fall Acceleration
Free Fall Acceleration:
Falling Objects:
In the absence of air resistance, all objects fall with the same acceleration, although this may be hard to tell by testing in an environment where there is air resistance.
The picture shows an apple and a feather falling in vacuum with identical motions.
Figure 2-12

Free-Fall Acceleration

Free-fall acceleration
 is the rate at which an object accelerates downward in the absence of air resistance o o o o
Written as g , standard value of 9.8 m/s 2
Independent of the properties of the object (mass, density, shape )
In vertical flight (along the y axis)
Where air resistance can be neglected

Free falling
•
Free falling with constant acceleration g = 9.8 m/s 2
•
The equations that represent motion of any object at y-axis (free falling) :
© 2014 John Wiley & Sons, Inc. All rights reserved.

Free falling
Example :
( g = 10 )
A ball is thrown vertically upward with a speed 18 m/s.
Find: a . The maximum height ?
b . Total time of flight ?
© 2014 John Wiley & Sons, Inc. All rights reserved.

Free falling solve: a . At The maximum height the final speed of the ball will be zero
V 2 f
=V 2 i
- 2g(y f
– y i
)
0 = (18) 2 – 2(10)Y f
Y f
= 16.2 m b . Total time of flight is the time needed to reach the maximum height
×
2
V
F
= V i
– gt ͢
0 = 18 – (10) t t = 1.8 s ͢ T total
= 3.2 s
© 2014 John Wiley & Sons, Inc. All rights reserved.

Free falling
Example :
A ball is thrown vertically upward with a speed of 25 m/s
Find its position after 2 s .(g = 10) solve:
Y
F
= Y i
+ V i t – ½ gt 2
Y
F
= 0 + (25)(2) – ½ (10)(4)
Y
F
= 30 m
© 2014 John Wiley & Sons, Inc. All rights reserved.

Free-Fall Acceleration
Example 1: A stone is dropped from rest from the top of a building, as shown in the Figure.
After 3s of free fall, what is the displacement y of the stone?
Solution:
From equation 𝑦 𝑓
= 𝑦 𝑖
+ 𝑣 𝑥𝑖
.𝑡 − 1
2
.𝑔.𝑡 2 𝑦 𝑓
= 0 + 0 −
1
2
. 9.8 . 3 2 = −44.1 𝑚

Free falling
HW # (g = 10 )
A stone is freely dropped from a height of 120 m above the ground . Find
•
Its speed just before hitting the ground . answer (V
F
= -49 m/s down)
•
The time taken for this motion . answer ( t = 4.9 s)
© 2014 John Wiley & Sons, Inc. All rights reserved.



Position
Relative to origin
Positive and negative directions
Summary

Displacement
Change in position (vector)
Eq. (2-1)

Average Velocity
Displacement / time (vector)

Average Speed
Distance traveled / time
Eq. (2-2)
Eq. (2-3)

Summary


Instantaneous Velocity
At a moment in time
Speed is its magnitude
Eq. (2-4)

Average Acceleration
Ratio of change in velocity to change in time
Eq. (2-7)


Instantaneous Acceleration
First derivative of velocity
Second derivative of position

Constant Acceleration
Includes free-fall, where
a = -g along the vertical axis
Eq. (2-8)
Tab. (2-1)

HW
Problems : page 41 - 44
5 , 14, 15 , 25 , 26 , 27 ,28 , 33 , 60 , 64
© 2014 John Wiley & Sons, Inc. All rights reserved.