Download Lecture - Vectors File

SACE Stage 1
Conceptual Physics
Vectors
Vector and Scalar Quantities


Quantities that require both magnitude and
direction are called vector quantities.
Examples of vectors are Force, Velocity
and Displacement.
Vector and Scalar Quantities


Quantities that require just magnitude are
known as Scalar quantities.
Examples of scalar quantities are Mass,
Volume and Time.
Vector Representation of Force

Force has both magnitude and direction
and therefore can be represented as a
vector.
Vector Representation of Force

The figure on the left shows 2 forces in the same
direction therefore the forces add. The figure on
the right shows the man pulling in the opposite
direction as the cart and forces are subtracted.
Vector Representation of Velocity


The figure on the left shows
the addition of the wind speed
and velocity of the plane.
The figure on the right shows
a plane flying into the wind
therefore the velocities are
subtracted.
Vector Representation of Velocity
Vector Representation of Velocity
Geometric Addition of Vectors

Consider a pair of horses pulling on a boat.

The resultant force is the addition of the
two separate forces F1 + F2.
Geometric Addition of Vectors
The resultant vector
(black) is the
addition of the other
2 vectors (blue +
green)
Mathematical Addition of Vectors

When we add vectors mathematically, we
use a vector diagram. This may include
using Pythagoras’ Theorem.
Mathematical Addition of Vectors

Pythagoras’ Theorem, in a right angled
triangle, the square of the hypotenuse is
equal to the sum of the squares of the other
two sides.
a2 + b2 = c2
Mathematical Addition of Vectors

Example – An 80km/hr plane flying in a
60km/hr cross wind. What is the planes
speed relative to the ground.
Mathematical Addition of Vectors

Solution
Use Pythagoras’
Theorem to find R
By Pythagoras ' Theorem 
R 2  ( 80 )2  ( 60 )2
 R 2  6400  3600
Draw a vector representation
of the velocities involved.
 R 2  10000
 R  10000
 R  100km / hr
Mathematical Addition of Vectors
As velocity is a vector, we need
to find the direction of the vector.
Can do this by finding an angle
(a) with in the vector diagram.
Use trigonometry to find the
angle.
opposite
adjacent
60
 tan a 
80
60
a  tan 1 ( )
80
a  36.9
tan a 
Mathematical Addition of Vectors

The answer should include both the size
and direction of the vector.
The velocity of the plane relative to the ground is
100km/hr at 36.9o to the right of the planes initial
velocity.
Equilibrium

Combining vectors using the parallelogram
rule can be shown by considering the case
of being able to hang from a clothes line
but unable to do so when it is strung
horizontally, it breaks!
Equilibrium


Can see what happens when
we use the spring scales to
measure weight.
Consider a block that weighs
10N (1Kg), if suspended by a
single scale it reads 10N.
Equilibrium

If we hang the same block by
2 scales, they each read 5N.
The scales pull up with a
combined force of 10N.
Equilibrium

What if the 2 scales weren’t vertical but
were attached at an angle. We can see for
the forces to balance, the scales must give
a reading of a larger amount.
Components of Vectors

The force applied to
the lawn mower may
be resolved into two
components, x for
the horizontal and y
for the vertical.
Components of Vectors


The rule for finding the vertical and
horizontal components is simple.
A vector is drawn in the proper direction
and then horizontal and vertical vectors are
drawn from the tail of the vector.
Components of Weight


Why does a ball move faster on a steeper
slope?
We can see what happens when we
resolve the vector representing weight into
its components.
Components of Weight


Vector A represents the amount of
acceleration of the ball and vector B
presses it against the surface.
Steeper the slope, more A.
Projectile Motion


A projectile is any object that is projected
by some means and continues in motion by
its own inertia.
An example is a cannon ball shot out of a
cannon or a stone thrown in the air.
Projectile Motion

The horizontal component of the motion is
just like looking at the horizontal motion of
a ball rolling freely on a horizontal surface.
Projectile Motion

The vertical component of an
object following a curved path is
the same as the motion of a
freely falling object as discussed
in section 2.
Projectile Motion

A multi-image
photograph
displaying the
components of
projectile motion.
Projectile Motion

The horizontal component of the motion is
completely independent of the vertical
motion of the object and can be treated
differently.
Ph14e – projectile motion
Projectile Motion

In summary, the a projectile will accelerate
(change its speed) in the vertical direction
while moving with a constant horizontal
speed. This path is called a parabola.
Upwardly Moving Projectiles

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Imagine a cannon ball shot at an upward
angle in a gravity free region on Earth. The
cannon ball would follow a straight line.
But there is gravity, the distance the
cannon ball deviates from the straight line
is the same distance that is calculated from
a freely falling object.
Upwardly Moving Projectiles
Upwardly Moving Projectiles

The distance from the dotted line can be
calculated using the formula introduced
previously.
1 2
d  gt
2
Upwardly Moving Projectiles

The following diagram shows the vectors
that represent the motion of the projectile.
Only the vertical component is
changing, the horizontal
component has remained the
same.
Upwardly Moving Projectiles

The horizontal
component of the
motion will determine
the range (how far
horizontally the
projectile will travel).
Upwardly Moving Projectiles

The following diagram displays the different
angle of a projectile launched with the
same initial speed.
Upwardly Moving Projectiles

Angles that add up to 90 degrees and
launched with the same initial speed have
the same Range.
Ph14e – projectile motion
Air Resistance on a Projectile

Air resistance affects both the horizontal
and vertical components of the motion
negatively.
Air Resistance on a Projectile


Need to consider how
air resistance effects
the horizontal and
vertical motion
separately.
Continuously slows
down horizontally and
maximum height is
reduced.
Physics in Surfing