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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