Excerpts of Chapter 1 and Chapter 3

Vectors and Two Dimensional Motion

• Vectors

• Operations with Vectors – Graphical Method

• Operations with Vectors – Component Method

• Two Dimensional Motion: Projectile Motion

• Kinematic Model

• Flight time, Range, Maximum Height

• Examples

That is, describing how objects

move in a plane

Vectors – Properties

Scalars are physical quantities exclusively described by numerical values

Ex: time, mass, temperature, etc.

Vectors describe physical quantities having both magnitude and direction.

Ex: position, displacement, velocity, acceleration, force, etc.

magnitude

θ

θ or direction

direction

• The direction of a vector depends on the arbitrary system of coordinates

• However, the magnitude does not depend on how you choose to span the space

V

V

Vor

y

x

y

x Angle in standard position

Operations with Vectors – Multiplication with a number

• A set of vectors can be added or subtracted in any order.

• If the vectors added represent physical quantities, they must have the same nature.

• Multiplying a vector by a positive number multiplies its magnitude by that number

(if the number is negative the vector flips in the opposite direction):

V

2V

2V

2V

2V

V

Operations with Vectors – We’ve already encountered them…

• In order to add or subtract 1D vectors, it suffices to add the vectors algebraically:

if the signs are correctly specified, the resultant vector will have the correct

direction and magnitude

1x 2x

1 2 10 km 4 km 14 kmx x

One dimensional case:

• Position, displacement, velocity and acceleration are all vectors

• In the 1D case they are confined along a straight line, so their direction is given by

the sign: positive x or negative x

x v

a0 x

Particle

Ex: The total displacement is the sum of the successive displacements

0

x (km)

10 km 4 km

Resultant = 14 km (east)

East 1x

2x

0

x (km)

Resultant =

6 km (east)

East

1 2 10 km 4 km 6 kmx x

• In general, even if the vectors are not along the same axis, they can be added

graphically by using the “tail-to-tip” method:

Operations with Vectors – Graphical addition: tail to tip method

• The method offers a qualitative idea about the resultant: in order to obtain the

resultant numerically (magnitude and direction), one has to use scaled grid paper

which is a rather cumbersome technique

The vector sum can be obtained graphically by chaining the vectors each with the

tail to the tip of the previous: then the vector resultant connects the tail of the first

vector to the tip of the last one. The operation can be done in any order.

Ex: Say that we have 3 arrows (vectors) in a plane (2D) and we want to add them up:

1v + 2v 3v+ = R

1v2v

3v

1 2 3R v v v

Notice that in 2D, the arrows above the vector symbols cannot be skipped since a vector

can have an infinity of directions not only two as in the 1D case: the operation between the

arrows cannot be reduced to an immediate algebraic addition or subtraction

Ex: Physical example: Successive 2D displacements can still be added to obtain the

total displacement

initial final

netd

1d

2d

3d

• An application of vector

summation in mechanics is

calculating the net

displacement of an object

traveling from an initial

position to a final one via

several successive partial

displacements

• If we denote d1, d2 and d3 three

successive displacements the

net displacement is

• It is given by the vector sum

(or resultant) of the partial

displacements

1 2 3netd d d d

• Note that adding the partial displacement follows the logic of tail-to-tip method

Operations with Vectors – Subtraction

• In order to subtract vectors, we can still use the addition procedure by adding the

negative of the arrow being subtracted

• We define the negative of a vector to be a vector with

the same magnitude but pointing in the opposite

direction.

v v

1v

_

2v

= R

1v

2v

1 2 1 2R v v v v

Ex: Say that we have 2 arrows (vectors) in a plane and we want to subtract them:

=

1v

+

2v

Ex: Physical example: linear displacement is defined as the final position minus the initial

position

reference

2 1r r r

initial

1r

final

2r

r

2 1r r r

• If we denote r1 and r2 two

positions successively

occupied by a moving

objects, the displacement is

Vectors – Components

• Note that, in order to obtain magnitudes and directions, the graphical methods

should be used on grid paper.

• A more computational way to get magnitudes and directions is by using vector

components in arbitrary systems of coordinates:

y

x

V

xV

yV

2 2

1

cos Components from

sin direction and magnitude

Direction and magnitude

from componentstan

x

y

x y

y

x

V V

V V

V V V

V

V

θ

Notation: ,x yV V V

Caution: The components are not are not vectors or vector magnitudes. They can be

negative if the corresponding vector components point in the negative direction of

the respective axis.

Operations with Vectors – Addition using vector components

• The addition and subtraction of vectors can be reduced to the algebraic addition

and subtractions of components

• Given n vectors in the same plane, the addition can be solved in 2D as following:

1 2

1 2

1 2

......

...

x x

y y

x

y

V VV V

V V

RR

R

2 2

x yRR R

1tan y xR R

magnitude:

direction:

1. Calculate the components of the

resultant by adding the vector

components in the respective

directions:

2. Calculate the magnitude and direction

of the resultant from its components:

Ex: The procedure can be visualized graphically:

the components (Rx, Ry) of the resultant R are

aligned with the components of the vectors

involved so they can be added as numbers

R A B

Problem:

1. Operating with vectors: Given the two vectors in the figure, find the vector resultants

where and are vectors with magnitudes 4 and 5 units respectively, by using

a) Graphical method

b) Vector components

1 2R A B

A

B

30

4

5

A B

2 2R A B

Projectile Motion – Main idea

• A typical example of 2D motion is the a generic free fall: the motion of a projectile

launched with an arbitrary initial velocity while only its weight acts on it (so, we

neglect air resistance)

• One way to model this motion is to project it along vertical and horizontal axes:

then the velocity in the x-direction is constant while in the y-direction the

kinematics is with constant acceleration 𝒈

Ex.

• If two objects are released simultaneously – one vertically

down and one with an initial horizontal velocity – the

vertical motion (red ball) is the same as the vertical

projection of the parabolic motion (yellow ball), like the

motion of a shadow on a vertical screen

• Consequently, the yellow ball will stay at the same

altitude as the red ball at all times

Projectile motion – Systematic Approach

• The acceleration of the projectile is only the gravitational acceleration g pointing

vertically downward: so only the y-component of the motion is accelerated

• In turn, the x-component is at constant velocity

y

x Projectile

0v

trajectory

000 sinvv y

000 cosvv x

, 0,x ya g a a g

0x

y

a

a g

Ground

t0=0

t

Time t

y-components:

210 2yy v t gt

0y yv v gt

Time t0

Initial velocity

yv

xv

v

Time t

x-components: 0x xv v0xx v t

Δy (t)

= y ‒ y0

Δx (t) = x ‒ x0 0

r

y

x

r

x (t)

y (t)

v

vx

vy

Exercise 1 Free falling objects: Consider again the two

objects released simultaneously. Using the logic on the

previous slide, the kinematic vectors for the yellow ball are:

0

2120 0 0, ,x yy vr x y x v gtt t

Position:

Velocity:

Acceleration:

00, ,x x yyv v v v v gt

, 0,x xa a a g

a) What is its approximate speed of the red ball 0.4 seconds

after release?

A) ~10 ms B) ~4 m

s C) ~8 m

s

The y-components

match the equations

for the red ball

b) Say that the initial speed of the yellow ball is 3 m/s. What

is its approximate speed 0.4 seconds later?

0 yv

Total time ttot, Range R, Maximum height ymax

x

0

0v

0

R

ymax

y

0xv

Problem:

2. Trajectory characteristics: Demonstrate the following relationships for the total time ttot,

the horizontal range R, and maximum height ymax on the trajectory shown above:

0 02 sintot

vt

g

2

0 0sin 2vR

g

2 2

0 0sin

2max

vy

g

Problem

3. Hitting a target: A plane flies horizontally with a

constant speed v at an altitude h above the ground. Say

that a melon is to be dropped to hit a smug Kim

Kardashian who does whatever it is that she does

somewhere on the ground below. Neglecting air

resistance, find an expression in terms of v, h and g for

the horizontal distance d ahead of Kim where the melon

is supposed to be dropped in order to make things right?

v

h

d

Exercise 2: Bad Physics in “Pearl Harbor”. Take for instance the 2001 war drama Pearl

Harbor which is marred not only by historical inaccuracies, but also by completely bogus

airplane maneuvers and bomb trajectories. In the adjacent clip you can follow the trajectory of

a bomb that hits USS Arizona apparently after being dropped from an airplane almost

vertically above the ill-fated deck. Let’s analyze it:

When it comes to Physics, if ignorance is bliss, Hollywood is the garden of Eden. Their movies are a

bottomless fountain of references to how subculture seems to offer more respect to the rules of baseball

than to the laws of nature…

Knowing that Arizona was bombed from about 3000 m altitude and the service speed of a B5N

Kate bomber is about 300 km/h, estimate the horizontal distance before the battleship where

the bomb must have been dropped in order to hit it (neglecting air drag).