Ch.3Scalars & Vectors

Scalar: e.g.

Vector: e.g.

Vector Notation: using vector A.A or A

(text books – bold) (writing on paper)

On paper, vectors are represented as with magnitude (size) and direction.

25m/s 250m

E 45o

Adding VectorsVectors can only be added if they are in

the

e.g. velocity + velocity, acc. + acc.but NOT velocity + acc.

Vectors are added via the Method or the

method.When adding vectors, the overall result is

known as the Vector. (R)

i.e. R = A + B aka: Nett vector

The Triangle MethodIn this method, the vector arrows are

added .e.g. Diagrammatically show R = A + B

where A & B are:

A B 20m/s40m/s 45o

Need to discuss scale:So 40m/s could be drawn cm long.and the 20m/s would be cm long

@ 45o

- Solve graphically. (Use a BLOODY ruler! And a protractor)

Measure the length of R and the angle .Using the scale, convert the length to a

magnitude.So R is m/s @ o above the

horizontal axis.

Solve mathematically using:Law of Cosines & Law of Sines.

R 20m/s 135o 45o

40m/sc2 = a2 + b2 – 2abCos

SinB = SinC b c

The Parallelogram MethodIn this method, the vector arrows are

added , and a parallelogram is formed.

Use the previous example and solve again.

Note: R = A + B is the same as

R = B + ASo it doesn’t matter which arrow you

start off with.

Referencing DirectionWhen stating the direction, you

need to include ane.g.

Vector B is 20m/s @Or

Another option: Bearingsi.e. N, E, W, S etc…

B 20m/s

45o

E.g. A plane is flying from LA to Miami at a speed of 180.0m/s (~400mph) on an Easterly bearing. It encounters a crosswind of 45.00m/s directly South. What will be the plane’s true speed and direction?

Solve graphically & mathematically.180.0m/s East

R 45.00m/s

South

Negative VectorsA negative vector has the same

magnitude as the positive vector, but points in the direction.

A -A

B45o

Subtracting VectorsVectors can only be added. To subtract, you must .i.e. to solve:

R = A - B you must do:

R = A + (-B)

E.g. Solve R = A + B & R = A - B

where:

A B 20m/s

40m/s 45o

R = A + BR 135o B

A

R = A – B = A + (-B)

Multiplication/Division of Vectors

Multiplication/division of a vector increases/decreases the only of a vector by a given factor. (a scalar number).

The is NOT affected.E.g. 3B is 3 times the magnitude of

B but still in the same direction. 15m/s 45m/s

B 3BSo: Scalar x Vector = Vector (diff.

mag, same dir.)e.g. time x velocity = displacementWhat about Vector x Vector = ?

Vector ComponentsA vector can be represented by its x &

y components.E.g.

A 5m 30o

Can be represented by:5m

A 30o

So A =

We can now say:

Cosθ = Ax

A

Sinθ = Ay

A

Tanθ =

A2 =

Projectile MotionWhen an object is launched into the

air, it is solely under the influence of . If this projectile is launched at an angle, it will follow a path in an shape. This is known as Projectile Motion.

θAssumptions:i.) vx is constant because ax = 0 (no

drag)ii.) ay = -g = -9.80m/s2

iii.) vy = 0 @ the highest pt (Apex, Zenith…)

What happens to each of the components during the flight?

vyo vo

θ vxo

vxo =

vyo =

Projectile Motion Equations

x-direction y-direction

vxo = vo.Cosθ vyo = vo.Sinθ

x = vxo.t vyf = vyo – gt

y = vyo.t – ½ gt2

vyf2 = vyo

2 – 2gy

Question: Which will take longer to hit the ground: A ball that rolls off the edge of a table, or a ball that is launched horizontally off a table?

Answer: They both fall at the same rate and hit the ground at the same time.

Example: A bird with a worm in it’s beak is flying horizontally at 5m/s at a height of 25m. The worm wriggles itself free. From the moment of release,

(a) how long will it take for the worm to hit the ground, and

(b) how far along the ground will it land?

A fireworks rocket is launched at an angle

of 25o to the ground. If it takes off at a speed of 12.0m/s, then:

a.) How long till it lands,b.) How far away will it land, andc.) What maximum height will it reach?

Example: Robbie Knievel “the DareDevil” is attempting another stunt. He will ride his motorcycle at 30m/s up a 32o angle ramp that reaches a height of 20m.

(a) What is the maximum height a wall can be for him to still clear it,

(b) The total time in the air(c) How far along will he land, and(d) What velocity will he land at?

20m

x

30m/s32o

vo = 30m/s

= 32o h = 20m a.) hmax = ?

b.) tTotal = ?

c.) x = d.) vf = ?

vxo =

= =

vyo =

= =

vxf

vyf

vf

Challenge question:What angle will achieve the greatest

distance in the x direction?Ans: = o (Could you prove

it?)

Note: If A + B = o, thenx from angle A = x from angle B