SACE Stage 1 Conceptual Physics Vectors. Vector and Scalar Quantities Quantities that require both...

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

Draw a vector representation of the velocities involved.

Use Pythagoras’ Theorem to find R

hrkmR

R

R

R

)()(R

/100

10000

10000

36006400

6080

Theorem 'PythagorasBy

2

2

222

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 () with in the vector diagram.

Use trigonometry to find the angle.

9.36

)80

60(tan

80

60tan

tan

1

adjacent

opposite

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 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.

2

2

1gtd

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