Motion Along a

Straight Line

In this chapter, all vectors have only one nonzero component,

frequently only an x component.

Measuring Position

Model of 1 – D Motion:Motion is a continuous

change of position with

time.

The object moves along a

line, which will be an axes

of a coordinate system.

Cartesian system (x – y) Position along the “y” direction:

Vertical motion/position uses the variable “y”

yi = initial “y” position, yf = final “y” position

Position along the “x” direction:

Horizontal motion/position uses the variable x.

xi = initial “x” position, xf = final “x” position

+ direction

– direction

– x coordinates

– y coordinates

+ y coordinates

+ x coordinates

Measuring Position

Displacement is a vector quantity that measures an objects change in position from the origin.

Units: Metres (m)∆x = x2 − x1 ∆y = y2 − y1 (2.2)

"∆" (delta) is a Greek letter used to represent a change in any quantity.

Example:

∆𝑡 = 𝑡2 − 𝑡1 this denotes a final time minus an initial time.

Distance is a scalar that

measures the length of a path

taken from one position to

another.

Units: Metres (m)

Distance cannot be negative

because it can only get larger

Measuring Position

Suppose you kick a football in a straight line along the ground from 1

metre to 7 metres.

If the football were kicked from 7.0 m to

0.0 m the displacement would be

negative.

∆𝑥 = 𝑥2 − 𝑥1 = 7𝑚 − 0𝑚 = −7𝑚

Note the conditions for which

displacement and distance can be

equal.

A measure of the rate of change of

position is referred to as speed or

velocity.

This is a displacement in the positive

direction.

∆𝑥 = 𝑥2 − 𝑥1 = 7𝑚 − 1𝑚 = 6𝑚

x1 x2

Please note that the terms

”speed” and “velocity” are

NOT interchangeable!

Motion Average Velocity is a vector quantity that represents the

average value for the rate an object changes position in a

time interval.

The sign of Velocity indicates the direction in which the object

moves (positive or negative direction).

Because this is an averaged value, it can also be equated

using the initial (starting) speed and final speed.

Average Speed is a scalar quantity equal to the total

distance traveled divided by the time interval.

Speed is a scalar and cannot have a negative sign

𝑣𝑎𝑣𝑔,𝑥 =∆𝑥

∆𝑡=

𝑥2 − 𝑥1

𝑡2 − 𝑡1𝑣𝑎𝑣𝑔,𝑦 =

∆𝑦

∆𝑡=

𝑦2 − 𝑦1

𝑡2 − 𝑡1(2.3)

𝒗𝒂𝒗𝒈 =𝑣2 + 𝑣1

𝟐

Example 2.1 A swimmer swims, in a pool that is 50.0 m long. She swims a

length at racing speed, taking 24.0 seconds to cover the

length of the pool. She then takes twice that time to swim

casually back to her starting point. Find

a) Her average velocity for each length?

b) Her average velocity for the entire swim?

x2 = 277 m

start

x1 = 0 mx2 = 50 m

50 𝑚

Modeling Motion with Graphs In math terms, we refer to this as position is a function of time written

as 𝑥(𝑡).

Position is plotted along the vertical and Time along the

horizontal.

The Average Velocity is the slope of the secant line joining two

points on a graph of 𝑥(𝑡).

A secant line is a straight

line joining two points on a

function. It is equivalent to

the slope between two

points.

How can this be 𝑉𝑎𝑣𝑔 ?

Consider the equation:

𝑣𝑎𝑣𝑔,𝑥 =𝑥𝐵 − 𝑥𝐴

𝑡𝐵 − 𝑡𝐴

This is the equation for

slope!

Position as a Function of Time Note; the curve does not

show the path of an object!

Compare the slope of the

blue secant line to the

green.

Blue’s slope is greater

and thus the vavg is

greater as well!

c

Interpreting the 𝑥(𝑡) plot:

A curved line on a 𝑥(𝑡) graph indicates a changing

velocity.

A straight (non-horizontal) line indicates a constant

velocity.

A horizontal line (slope = 0) indicated a velocity of zero.

Instantaneous Velocity Instantaneous Velocity is an objects velocity at any one

specific instant of time.

This is still velocity; however, it is not an average value.

0

100

200

300

0 1 2 3 4

Po

sitio

n (

m)

Time (s)

𝐵

𝐴

The Instantaneous Velocity is

the slope of the tangent line

to the curve at a given point

of a 𝑥(𝑡) graph.

This is defined as a line

through a pair of infinitely close points on the curve.

𝑣𝑥 = lim∆𝑡 →0

∆𝑥

∆𝑡

• x tells use where the particle is and vx tells us how it is moving.

x2

x1

t1 t2

Position Vs TimeA) Has a positive slope

B) Has a positive slope

greater than that at

point A.

C) Has a slope of zero

D) Negative slope

E) Negative slope less

than that at point D

Po

sitio

n (

m)

Time (s)

A

B

C

D

E

Example 2.2

A cheetah is crouched in ambush 20.0 m to the east of an

observer’s vehicle. At time t = 0, the cheetah charges an antelope in

a clearing 50.0 m east of the observer. The cheetah runs along a

straight line; the observer estimates that, during the first 2.00 s of the

attack, the cheetah’s coordinate x varies with time t according to

the equation.

𝑥 = 20.0 𝑚 + 5.00𝑚

𝑠2𝑡2

a) Find the displacement of the cheetah during the interval

between t1 = 1.00 s and t2 = 2.00 s.

b) Find the average velocity during this time interval.

c) Estimate the instantaneous velocity at time t1 = 1.00 s by taking ∆𝑡 = 0.10 s.

Average and Instantaneous Acceleration

When the velocity of an object changes with time, we say that

the object has acceleration 𝑎.

𝑎 is the rate of change of velocity with time.

𝑎 is a vector quantity

Units: m/s2

Average Acceleration is the average value of acceleration

over a time interval.

𝑎𝑎𝑣𝑔,𝑥 =𝑣2𝑥−𝑣1𝑥

𝑡2−𝑡1(2.5)

Instantaneous acceleration is the acceleration of a body at a

specific instant in time.

Example 2.3

An astronaut has left the space shuttle on a tether to test a

new personal maneuvering device. She moves along a straight

line directly away from the shuttle. Her onboard partner

measures her velocity before and after certain maneuvers,

and obtains the following results:

If t1 = 2 s and t2 = 4 s in each case, find the average acceleration

for each set of data.

a) V1x = 0.8 m/s V2x = 1.2 m/s (speeding up)

b) V1x = 1.6 m/s V2x = 1.2 m/s (slowing down)

c) V1x = - 0.4 m/s V2x = -1.0 m/s (speeding up)

d)V1x = -1.6 m/s V2x = - 0.8 m/s (slowing down)

Instantaneous Acceleration Instantaneous Acceleration follows the same procedure we

use for velocity.

Plotting a graph with velocity vx on the vertical axis and time t on

the horizontal axis. The Average acceleration is the

slope of the secant line joining

two points on a graph of velocity

as a function of Time.

Again consider the equation:

𝑎𝑎𝑣𝑔,𝑥 =𝑣2 − 𝑣1

𝑡2 − 𝑡1

The Instantaneous Acceleration

is the slope of the tangent line to

the curve at a given point of a

velocity as a function of time.

Speeding up Slowing down

+ vx and + ax ( In the + dir.) + vx and - ax ( In the + dir.)

- vx and - ax ( In the - dir.) - vx and + ax ( In the - dir.)

Example 2.4

Suppose that at any time t, the velocity v of the car below is given by the equation:

𝑣𝑥 = 60.0𝑚

𝑠+ 0.5

𝑚

𝑠3𝑡2

a) Find the change in velocity of the car in the time interval between t1 = 1.00 s and t2 = 3.00 s.

b) Find the average acceleration in this time interval.

c) Estimate the instantaneous acceleration at time t1 = 1.00 s by taking ∆𝑡 = 0.10 𝑠

Motion with Constant

Acceleration

Velocity as a function of Time In this type of motion velocity changes at the same rate

throughout a certain time interval.

𝑎 is not changing, so it is a constant (instantaneous)

acceleration 𝑎𝑥

𝑎𝑥 =𝑣2𝑥 − 𝑣1𝑥

𝑡2 − 𝑡1

∴ 𝑎𝑥 =𝑣2𝑥 − 𝑣1𝑥

𝑡 − 0

∴ 𝑎𝑥 =𝑣𝑓𝑥 − 𝑣0𝑥

𝑡

Where, 𝑎𝑥𝑡 = ∆𝑣𝑥

Redefine variables:

Let 𝑡1 = 0𝑠 and 𝑡2 = 𝑡 (Any later time)

Let 𝑣0𝑥 be the velocity at 𝑡1 = 0𝑠

Let 𝑣𝑓𝑥 be the velocity at 𝑡2 = 𝑡

We obtain the equation of velocity

as a function of time or 𝑣(𝑡).

𝑣𝑓𝑥 = 𝑣0𝑥 + 𝑎𝑥𝑡 (2.8)

𝑣𝑓𝑥 = 𝑣0𝑥 + ∆𝑣𝑥

Example 2.5 A car initially traveling along a straight stretch of highway at 20

m/s accelerates with a constant acceleration of 2.5 m/s2 in

order to pass a truck. What is the velocity of the car after 7.5

seconds?

Position as a function of Time Under the same assumptions we can derive an equation for

position as a function of time 𝑥(𝑡).

The equation of position as a function of

time or 𝑥(𝑡).

𝑥 = 𝑥0 + 𝑣0𝑥 𝑡 +1

2𝑎𝑥𝑡

2(2.12)

This equation states that at an initial time (t = 0), a particle is at a

position of x0 and has a velocity v0x

Example(s) 2.5 - 2.6

a car travelling along a straight highway

at 15 m/s accelerates at 2.0 m/s to pass

a truck.

What is its velocity after 5.0 seconds?

What distance does the car travel during its

5.0 seconds of acceleration?

Velocity as a function of position

Sometimes a time interval isn’t given for the motion, and we

need to obtain a relation for x, vx and ax that doesn’t contain t.

The equation for velocity as a function of

position or 𝑣(𝑥).

𝑣𝑥2 = 𝑣0𝑥

2 + 2𝑎𝑥 𝑥 − 𝑥0 (2.13)

This equation gives us the particle’s velocity vx at any position x

without needing to know the time when it is at that position.

Example 2.7 A sports car is sitting at rest in a freeway entrance ramp.

The driver sees a break in the traffic and floors the car’s

accelerator, so that the car accelerates at a constant 10

m/s2 as it moves in a straight line onto the freeway. What

distance does the car travel in reaching a freeway speed

of 45 m/s?

Example 2.8 A sports car is sitting at rest in a freeway entrance ramp.

The driver sees a break in the traffic and floors the car’s

accelerator, so that the car accelerates at a constant 10

m/s2 as it moves in a straight line onto the freeway. What

distance does the car travel in reaching a freeway speed

of 45 m/s?

Position, Velocity and Time Sometimes a time interval isn’t given for the motion, and we

need to obtain a relation for x, vx and ax that doesn’t contain t.

The equation for velocity as a function of

position or 𝑣(𝑥).

𝑣𝑥2 = 𝑣0𝑥

2 + 2𝑎𝑥 𝑥 − 𝑥0 (2.13)

This equation gives us the particle’s velocity vx at any position x

without needing to know the time when it is at that position.

Objects Falling Freely A very common observation is that of an object falling under the

influence of Earth’s Gravitational attraction.

This is called free fall which includes rising objects as well as falling

motion.

Objects fall with an acceleration independent of their weight.

When air resistance is absent all objects at a particular location

fall at the same acceleration (g).

This is a vector quantity whose magnitude is 9.81 m/s2

The direction of this vector is downwards

As we analyze free fall we will use an idealized model in which

we neglect: air resistance, Earths rotation, and altitude.