Vectors and Two Dimensional
Motion
Vectors and their proprieties
• Every Physical quantity can be categorized
either a vector quantity or a scalar quantity
• A vector has: - direction and magnitude
• A scalar: - magnitude
• Two vectors A and B are equal if they have the
same magnitude and the same direction ( this
propriety allows us to translate a vector parallel
to itself in a diagram without affecting the vector)
Adding vectorsgeometrically
- when two ore more vectors are added
they must all have the same units !!!!
- Triangle method of addition to add vector
B to A geometrically first draw vector A ,
then draw B to the same scale with the tail
of B starting at the tip of A. The resultant
vector R =A +B is the vector drawn from
the tail of A to the tip of B.
- Commutative law of addition:
A + B = B +A
- Negative of a vector A is defined as the
vector that gives zero when added to A :
- They have same magnitude
but opposite directions!!
- Subtracting Vectors :
A – B = A +(- B)
- Multiplying or Dividing a Vector by a scalar
A multiply with 3 = 3 A is a vector with
magnitude three times that of A and
pointing in the same direction
(what happened if we multiply with -3, or
what happened if we divide with 3?)
Quiz 3.1 and 3.2/55
Components of a vector
Use the projections of a
vector along the axes
of a rectangular
coordinate system,
these projectors are
called components
A = A x + Ay
A x = A cosθ
A y = A sinθ
A = √ Ax2 +Ay2
tanθ = Ax /Ay
• If we choose the
coordinate differently,
the components of
the vector must be
modify accordingly
Bx’ = B cos θ
By’ = B sin θ
Quiz 33/ 57
Adding vectors algebraically
R = A +B
Rx = Ax + Bx
R y = Ay +By
Displacement, velocity and
acceleration in Two Dimension
- An object displacement is defined as the
change its position vector or
Δr = rf – ri (SI: m)
- An object’s average velocity during a time
interval Δt is its displacement divided by
Δt:
v av = Δr / Δt (SI: m/s}
- An object’s instantaneous velocity v is the
limit of its average velocity as Δt goes
zero: v = lim (Δt→0) Δr/Δt (SI: m/s)
- An object average acceleration during a
time interval at is the change in it's velocity
divided by Δt :
a av = Δv/Δt
(SI: m/s2)
- An object instantaneous acceleration is
the limit of its average acceleration vector
as Δt goes to zero
a = lim (Δt→0) Δv/Δt (SI: m/s2)
An important special case of this two dimensional
motion is called projectile motion.
The path of a projectile in Earth’s gravity field is
curved in shape of parabola
The horizontal and
vertical motions are
completely
independent of each
other
Relative Velocity
• The measured velocity of an object
depends of velocity of the observer
with respect to the object
• On highways cars moving in the same
direction are often moving at high speed
relative to Earth, but relative with each
other, they hardly move at all
• Measurements of velocity depend on the
reference frame (most of the time
reference to Earth)