Describing Motion: Kinematics

in One Dimension

Chapter 2

Definitions

Kinematics – branch of Physics that

describes how objects move

Translational motion – when an object

moves without rotating

Scalars vs. Vectors

Scalar – quantity that represents magnitude only (with units)

Magnitude – numerical quantity

Vector – quantity that represents magnitude (with units) and direction

Use arrows to represent

Length of arrow is proportional to magnitude

Tip of arrow indicates direction

VERY helpful to draw vector diagrams to work problems

Often have arrow drawn on top of symbol to be distinguished from scalars

Vectors can be positive or negative.

Think of a Cartesian plane.

X axis - Designate right as positive and

left as negative.

Y axis - Designate up as positive and

down as negative.

Vectors can be added graphically

Vectors must have the same units to do this.

Connect the tail of the first vector to the head of the last vector. This line is called the resultant.

For one dimensional motion, the resultant’s direction is the same as the vector with the largest magnitude.

To subtract vectors, add the opposite of the vector.

Distance and Displacement

Distance – total path covered by the

object from start to end (scalar)

Displacement – shortest path from start

to end (vector)

Which statement is true regarding

distance and displacement? A. Distance will never be greater than

displacement.

B. Displacement will never be greater than

distance.

C. Distance and displacement will always be

equal.

D. Distance and displacement will never be equal.

Displacement

Displacement

= change in position (x )

= final position – initial position (xf – xi)

x = xf – xi (horizontal motion)

y = yf – yi (vertical motion)

Example 1

A space shuttle takes off from FL and circles the Earth several times, finally landing in CA. While the shuttle is in flight, a photographer flies from FL to CA to take pictures of the astronauts as the step off the shuttle. Who has the greater displacement – the astronauts or photographer?

a. The astronauts

b. The photographer

c. They both have the same displacement.

d. There is not enough info to determine the answer.

Example 2

What is the coach’s distance travelled?

a. 0 yds e. -5 yds

b. 5 yds f. -55 yds

c. 55 yds g. -95 yds

d. 95 yds

Example 3

What is the skier’s displacement?

a. 40 m right e. 40 m left

b. 80 m right f. 80 m left

c. 100 m right g.100 m left

d. 140 m right h. 140 m left

Example 4

You start walking home from school. After walking

1.3 km North, you get a phone call on your cell

from your mom asking if you can meet her at

the mall. You will have to turn around and

walk 2.5 km South. Determine your distance

to get to the mall.

a. 1.2 km e. 1.2 km N i. 2.5 km N

b. 1.3 km f. 1.2 km S j. 2.5 km S

c. 2.5 km g. 1.3 km N k. 3.8 km N

d. 3.8 km h. 1.3 km S l. 3.8 km S

Example 4

You start walking home from school. After walking

1.3 km North, you get a phone call on your cell

from your mom asking if you can meet her at

the mall. You will have to turn around and

walk 2.5 km South. Determine your

displacement to get to the mall.

a. 1.2 km e. 1.2 km N i. 2.5 km N

b. 1.3 km f. 1.2 km S j. 2.5 km S

c. 2.5 km g. 1.3 km N k. 3.8 km N

d. 3.8 km h. 1.3 km S l. 3.8 km S

Speed

Scalar quantity that represents how fast

an object is moving

Rate at which an object covers a

distance (vavg = dtotal / ttotal)

Instantaneous speed measured by your

speedometer

Velocity

Vector quantity that represents the rate

at which an object changes position

(displacement / time)

vavg = x / t = xf-xi / tf-ti

SI Units = m/s and must include direction!

Example 1

Freddie walks 4 meters East, 2 meters South, 4 meters West, and finally 2 meters North. The entire motion lasted for 24 seconds. Determine the average speed.

a. 0 m/s e. 0.2 m/s W

b. 0.3 m/s f. 0.3 m/s N

c. 0.5 m/s g. 0.5 m/s E

d. 2 m/s h. 1 m/s S

Example 2

Freddie walks 4 meters East, 2 meters South, 4 meters West, and finally 2 meters North. The entire motion lasted for 24 seconds. Determine the average velocity.

a. 0 m/s e. 0.2 m/s W

b. 0.3 m/s f. 0.3 m/s N

c. 0.5 m/s g. 0.5 m/s E

d. 2 m/s h. 1 m/s S

Example 3

A car drives along the highway at 115

km/h for 2.50 h. Once in the city, the

car drives at 60.0 km/h for the next

0.500 h. Determine the average speed

of the car.

a. 58.3 km/h

b. 87.5 km/h

c. 106 km/h

Determining Velocity Graphically

Velocity can be determined from a

position-time or d-t graphs

Slope = y coordinates / x coordinates

= displacement /time

= m/s

= VELOCITY!

d-t graphs

+ y axis = positive direction

- y axis = negative direction

The x-axis represents your starting point.

Linear Slopes on d-t graphs

Horizontal line = displacement is zero (stationary)

Positive slope = moving at a constant velocity away

from start (+y-axis) or moving at a constant velocity

back toward start (-y-axis) = constant positive

velocity!

Negative slope = moving at a constant velocity

toward start (+y-axis) or moving at a constant velocity

away from starting point (-y-axis) = constant negative

velocity!

Example of a d-t graph

d-t graphs

Nonlinear graphs

When a graph is curved, the slope at each

point is the slope of a tangent line

Since each point has a different tangent,

curved d-t graphs show changing velocity

Non-linear Slopes on d-t graphs

A curve means the displacement is changing

at a non-constant rate (variable velocity)

If the steepness of the slope (tangent line)

increases, velocity is increasing in magnitude.

If the steepness of the slope (tangent line)

decreases, velocity is decreasing in

magnitude.

Nonlinear slope d-t graphs

Graphing Example

Describe the motion:

a. Stationary

b. Moving away from start at a constant velocity

c. Moving toward start at a constant velocity

d. Increasing velocity

e. Decreasing velocity

A

B C

D

E

F

Motion Diagram

Like a strobe photograph that shows

slow motion of object

Show displacement of object over equal

time intervals

If you walked in snow or mud…

What would the spacing of your footprints

look like if you were walking…

1. At a slow, constant pace

2. Then you started to speed up

3. Then slowed down until

4. Finally you stop?

Motion Map Parts

The dot: Indicates the position of the

object

The arrow (vector): indicates the

direction and speed of the object.

Dots and Arrows Together

Dot alone = not moving.

Dot and arrow together:

Position, direction and speed.

Direction and Size

Right = positive

direction

Left = negative

direction

The longer the

arrow, the greater

the velocity.

SLOW

FASTER

FASTEST

The grid….

Motion Maps are drawn along a grid to show

the position of the object.

Draw a minimum of 3 arrows to show a

pattern.

Forward, Constant Velocity, Slow

Forward, Constant Velocity, Faster

Motion Maps

Suppose that you took a stroboscopic picture of a car

moving to the right at constant velocity where each image

revealed the position of the car at one-second intervals.

What would it look like if the car were moving faster?

Series of Motion

•The object moves forward at constant velocity,

•then stops and remains in place for two seconds,

•then moves backward at a slower constant velocity.

Example 1

Draw a motion map.

The car is accelerating!

Each successive arrow

is longer, indicating the

velocity is increasing.

Example 2

Follow the arrows to

describe the motion…

Given the following motion diagram,

describe the object’s motion:

x0 x1 x2 x3 x4

Indicate position arrows (vectors)

1. The object is not moving.

2. The object is moving at a constant velocity.

3. The object is increasing speed.

4. The object is decreasing speed.

Sue runs towards Jim starting from rest. The dots in the motion

diagram below represent Sue’s position at 0.2-second time

intervals. Which diagram could represent Sue's initial motion?

Which motion diagram could represent Sue’s motion once she has

reached her maximum speed?

Acceleration

Vector quantity that represents a change in

VELOCITY of an object

Includes a change in speed or direction (or both)

May be positive or negative

Positive when

Increasing speed and traveling right or up (+/+)

Decreasing speed and traveling left or down (-/-)

Negative when

Decreasing speed and traveling right or up (-/+)

Increasing speed and traveling left or down (+/-)

a = v = vf – vi

t t

Units are in m/s2

When acceleration is constant,

Changes in position (velocity);

changes in velocity (acceleration)

x4-x3 (x1-x0)/t~v0 (x2-x1)/t~v1 x3-x2

In equal time intervals (t)

(visualizing acceleration can be difficult)

v1-v0 v2-v1 v3-v2

Velocity is shown by the arrows connecting two successive dots.

The following drawings indicate the motion of a ball from left to

right . Each circle represents the position of the ball at succeeding

instants of time. Each time interval between successive positions

is equal.

Rank each case from the highest to the lowest acceleration based

on the ball's motion using the coordinate system specified by the

dashed arrows in the figures. Note: Zero is greater than negative,

and ties are possible.

Acceleration Due to Gravity

Often called free fall acceleration

Because any object on Earth is so small compared to the size of Earth, we have a constant value for acceleration close to the Earth’s surface.

g = -10 m/s2: The value is negative because it is ALWAYS in the downward direction!

Used for value of a whenever there is vertical motion (up or down).

Free Fall Motion Diagram

Acceleration is more than just a formula –

Applying definitions, we get useful relationships

• vavg= x/t

• xf = xi + vavg t

• a=v/t

• vf = vi + a t

• If a is constant, then vavg = vf + vi

2

One Dimensional Motion Equations

Using the definitions of average velocity and

acceleration, we can derive 4 mathematical

equations for motion in one dimension.

x = ½ (vf + vi) t vf = vi + a t

x = vi t + ½ at2 vf2 = vi

2 + 2 ax

*Note: these formulas can only be used when an object

has a constant acceleration.

*For vertical motion, replace x with y and a with g

Determining Acceleration Graphically

Acceleration can be determined from a

velocity-time or v-t graph

Slope = y coordinates / x coordinates

= velocity /time

= m/s2

= ACCELERATION!

v-t graphs

+ y axis = positive direction

- y axis = negative direction

Crossing the x-axis means you changed

directions. It does not represent your

starting point!

Linear Slopes on v-t graphs

Horizontal line = constant velocity

Positive slope = increasing speed at a constant rate

in the positive direction or decreasing speed at a

constant rate in the negative direction (constant +

acceleration)

Negative slope = decreasing speed at a constant rate

in the positive direction or increasing speed at a

constant rate in the negative direction(constant –

acceleration)

Non-linear Slopes on v-t graphs

A curve means the velocity is changing

at a non-constant rate (variable

acceleration)

If the steepness of the slope increases,

acceleration is increasing in magnitude.

If the steepness of the slope decreases,

acceleration is decreasing in magnitude.

v-t graph example

What time interval(s) are in the

positive direction?

1. 0-3 s 4. 0-8 s

2. 0-5.5 s 5. 13-16.5 s

3. 0-5.5 s, 13-16.5 s

What time interval(s) are in the

negative direction?

1. 5.5 - 6.5 s 4. 5.5 – 13 s

2. 5.5 – 8 s 5. 8-13 s

3. 5.5 – 9 s 6. 8 – 16.5 s

What time interval(s) show positive

acceleration?

1. 0 - 3 s 4. 0 – 3, 13 – 16.5 s

2. 0 – 5.5 s 5. 9 - 16.5 s

3. 0 – 8 s 6. 13 – 16.5 s

What time interval(s) show negative

acceleration?

1. 3 – 5.5 s 4. 8 – 16.5 s

2. 5.5 - 8 s 5. 8 – 9, 13 - 16.5 s

3. 5.5 - 9 s

How to determine the distance and

displacement from a v-t graph

v t = x (so on this graph, that is y x, or length x width)

To find the area under the curve, separate into triangles and/or rectangles.

Calculate the area of each object and add them all together.

For distance, add all magnitudes together ignoring + or –

For displacement, you must consider the sign as + or – and then add together.

Area under the curve example

Find the displacement during these

55 seconds of motion