Motion in One Dimension

Chapter 3

Outline

3.1 Position, Velocity and Speed

3.2 Instantaneous Velocity and Speed

3.3 Acceleration

3.4 Motion Diagrams

3.5 One-Dimensional Motion with Constant Acceleration

3.6 Freely Falling Objects

3.1 Position, Velocity and Speed

Position Displacement

An object’s position is its

location with respect to a

chosen reference point

• the change in position during

some time interval

• represented by x

x ≡ xf – xi

x is positive when xf > xi

x is negative when xf < xi

Vectors and Scalars

A vector – physical quantity that require the specification of both

direction and magnitude

use “+” and “-” sign to indicate vector directions

ex : displacement, velocity and acceleration

Displacement in positive (+x) means object move to the right and (-

x) means move to left

• A scalar – A quantity that has magnitude but no direction

ex : position, speed and etc.

3.1 Position, Velocity and Speed cont.

Average speed Average velocity

Speed is a scalar quantity

Defined as

It has no direction, so always positive value.

• the displacement x occurs at the interval of time.

x indicates motion along x- axis

Average velocity of a moving particle can be positive and negative depends on the sign of displacement

• x is positive when xf > xi , Vx. avg is positive

x is negative when xf < xi , , Vx. avg is negative

3.1 Position, Velocity and Speed cont.

Average speed Average velocity

Distance traveled per unit of time Displacement traveled per unit of time

A

Bd = 30 m

Time t = 5 s

• Direction is not (scalar)

A

Bd = 30 m

Time t = 5 s

x =12 m

20o

V = 2.4 m/s at 200 N of E

•Direction required (vector)

3.2 Instantaneous Velocity and Speed

Instantaneous velocity can indicate what is happening at

every point of time.

General equation for instantaneous velocity :

Instantaneous velocity can be positive, negative or zero.

Instantaneous speed is the magnitude of the instantaneous

velocity but it has no direction.

0limx

t

x dxv

t dt

Instantaneous Velocity, graph

The instantaneous velocity is the

slope of the line tangent to the x

vs. t curve.

This would be the green line.

The light blue lines show that as

t gets smaller, they approach the

green line.

3.3 Acceleration

Acceleration is the velocity changes with time.

The average acceleration is the change in velocity Vx divided by

the time interval t during the change occurred.

“+” and “-” sign can indicate direction.

,x xf xi

x avg

f i

v v va

t t t

Dimension : L/T2

SI : m /s2

Instantaneous Acceleration the limit of the average acceleration as t approaches 0.

is also equals to the derivative of the velocity with respect to time.

acceleration is the slope of Vx vs t graph.

It has positive and negative value to indicate the direction.

2

20lim x xxt

v dv d xa

t dt dt

With :

Vx = dx/dt

Instantaneous Acceleration – graph

The slope of the velocity-time graph is the acceleration.

The green line represents the instantaneous acceleration.

The blue line is the average acceleration.

Ex 3.4 Average and Instaneous

AccelerationThe velocity of a particle moving along the x axis varies in time according to the expression

Vx = ( 40 – 5t2) m/s, where t is in seconds.

a) Find the average acceleration in the time interval t=0 to t=2s

b) Determine acceleration at t=2s

Notes about Velocity and Acceleration

Negative acceleration does not necessarily mean the object is

slowing down.

If the acceleration and velocity are in the same direction no

matter “+” / “-” , the object is speeding up.

If the acceleration and velocity are in the opposite direction,

the object is slowing down.

3.4 Motion Diagram

A motion diagram can be formed by imagining the stroboscope photograph of a moving object.

Red arrows represent velocity.

Purple arrows represent acceleration. All purple arrows maintain the same length means acceleration are constant.

Constant Velocity

The car image are equally spaced.

The car is moving with constant positive velocity. (red arrows

remain the same size)

Acceleration equals to zero.

Acceleration same direction with Velocity

The car image farther apart as time increases

Acceleration and velocity are in the same direction

Velocity is increasing as the arrows are getting longer

Acceleration is constant as the arrows maintain the same length

Thus, it is positive acceleration and positive velocity

Acceleration opposite direction with Velocity

The car image become closer as time increases

Acceleration and velocity are in the opposite direction

Velocity is decreasing as the arrows are getting shorter

Acceleration is constant as the arrows maintain the same length

Thus, it is negative acceleration and positive velocity

3.5 One-dimensional Motion with Constant

Acceleration

Constant acceleration.

Velocity changes at the same rate throughout the motion.

Kinematic Equation 1 :

xf xi xv v a t

Can determine the velocity at any

time t when we know its initial velocity

and its acceleration

Doesn’t tell about displacement

Kinematic Equation 2:

The average velocity in a time interval can be expressed as the

arithmetic mean of the initial velocity Vxi and final velocity Vxf

is only suit for constant acceleration.

,2

xi xfx avg

v vv

Kinematic Equation 3:

This gives us about the position of the object in terms of

time and velocities.

Doesn’t tell acceleration

Thus,

,2

xi xfx avg

v vv

Substitute Equation 2

then

Kinematic Equation 4:

This gives final position in terms of velocity and acceleration.

Doesn’t tell final velocity

Substitute Equation 3 with Equation 1

xf xi xv v a t

then 21

2f i xi xx x v t a t

Equation 3 :

Equation 1:

Kinematic Equation 5:

Tells final velocity in terms of acceleration and displacement.

Doesn’t include with time

xf xi xv v a t Equation 1: then

Equation 3 :

Substitute Equation 1 with Equation 3

2 2 2xf xi x f iv v a x x

When a = 0

When the acceleration is zero,

vxf = vxi = vx xf = xi + vx t

When acceleration is zero, velocity is constant and displacement

changes linearly with time.

Kinematic Equations - Summary

Example (3.7) : Carrier Landing

A jet lands on an aircraft carrier at 140 mi/h (63 m/s).

(a) What is its acceleration if it stops in 2.0 s?

(b) What is the displacement of the plane while it is stopping?

Example (3.8) : Watch Out for the Speed Limit!

A car traveling at a constant speed of 45.0 m/s passes a trooper hidden behind

a billboard. One second after the speeding car passes the billboard, the

trooper sets out from the billboard to catch it, accelerating at a constant

rate of 3.00 m/s2. How long does it take her to overtake the car?

3.6 Freely Falling Objects

A freely falling objects is any object that moves freely under

the influence of gravity alone

It doesn’t depend upon the initial motion of the object.

Thrown upward

Thrown downward

Dropped (release from rest)

• Any freely falling object experiences an acceleration directed

downward, regardless of the initial motion.

3.6 Freely Falling Objects cont.

The magnitude of free fall acceleration is symbol g

g decreases with increasing altitude

g varies with latitude

Define “up” as +y direction

the g value at Earth’s surface is 9.80 m/s2

• If we neglect air resistance, assume acceleration doesn’t vary with

altitude over distance, the free fall motion is constantly accelerated

motion in one dimension.

• While the motion in vertical direction and acceleration is downward

• we always take ay = -g = - 9.80 m/s2, the minus sign shows the acceleration of a

freely falling object is downward.

Free Fall – An Object Dropped

•Initial velocity, vo = 0

•Assume upward direction is positive

•Use the kinematic equations

– Generally use y instead of x since vertical

•Acceleration is

– ay = -g = -9.80 m/s2

vo= 0

a = -g

Free Fall – An Object Thrown Downward

•ay = -g = -9.80 m/s2

•Initial velocity 0

– With upward being

positive, initial velocity

will be negative.

vo≠ 0

a = -g

Section 2.7

Free Fall – Object Thrown Upward

•Initial velocity is upward, so

positive

•At the maximum height, The

instantaneous velocity is zero.

•ay = -g = -9.80 m/s2

everywhere in the motion

v = 0

vo≠ 0

a = -g

6.6 Sign Convention:A Ball Thrown Vertically

Upward

•• Velocity is positive (+) Velocity is positive (+) •• Velocity is positive (+) Velocity is positive (+) or negative (or negative (--) based ) based

on on direction of motiondirection of motion..

•• Displacement is positive Displacement is positive ) based ) based

•• Displacement is positive Displacement is positive (+) or negative ((+) or negative (--) based ) based

on on LOCATIONLOCATION. .

Release Point

UP = +

TippensTippens

•• Acceleration is (+) or (Acceleration is (+) or (--) ) based on direction of based on direction of forceforce(weight).(weight).

y = 0

y = +

y = +

y = +

y = 0

y = -

v = +

v = 0

v = -

v = -

v= -

a = -

a = a = --

Example (3.10) : Not a bad Throw

for a Rookie!

A stone thrown from the top of a

building is given an initial velocity

of 20.0 m/s straight upward. The

building is 50.0 m high, and the

stone just misses the edge of the

roof on its way down, as shown

in Figure (2.14). Using tA = 0 as

the time the stone leaves the

thrower’s hand at position ,

determine :

(a) The time at which the stone

reaches its maximum height,

(b) The maximum height,

(c) The time at which the stone

returns to the height from which it

was thrown,

(d) The velocity of the stone at this

instant, and

(e) The velocity and position of the

stone at t = 5.00 s