Transcript Topic 8 - Describing Straight Line Motion in 2D Space

TOPIC 8
Describing Straight Line Motion
in 2D Space
Elementary Vector Algebra in Two Dimensions
SCALARS are quantities such as mass, volume, distance, speed
and temperature which have a size but no associated direction.
VECTORS are quantities such as displacement, velocity, force
and acceleration which have both a size and an associated
direction.
A vector can be represented by a straight line whose length
reflects the magnitude of the vector and whose orientation
indicates the direction of the vector.
(a) Directed Line Segment - this is a vector joining two points
P
QP
Q
O
Elementary Vector Algebra in Two Dimensions
(b) Position Vector - this is a vector joining a point to the origin
P
p
O
The magnitude of a vector, a, is known as its MODULUS and
written as a
A UNIT VECTOR in a given direction is a vector with unit
magnitude in that direction.
The symbol i is used for the unit vector along the x axes and the
symbol j is used for the unit vector along the y axes. When we
write a vector using i and j it is said to be written in
COMPONENT FORM.
Elementary Vector Algebra in Two Dimensions
For example, consider the point L (4,7)
L
7
l = 4i + 7j
4
It has the associated position vector
l = 4i + 7j
To find its modulus we use Pythagoras’ Theorem
l = √x2 + y2
Definition of Modulus
= √42 + 72
= √16 + 49
l = √65
Elementary Vector Algebra in Two Dimensions
Example 1
5
(a) Change the column vector   into unit vector form.
 
8
(b)
Convert the vector -7j into column vector form
Answer 1
(a)  5 
  = 5i + 8j
8
 
(b) -7j =  0 
 7
 
Elementary Vector Algebra in Two Dimensions
Example 2
Given a = 3i + 2j and b = 5i – 6j find the resultant of a and b.
Answer 2
a + b = 3i + 2j + 5i – 6j
= 8i - 4j
Example 3
If p = 2i – 8j and q = -9i + j work out 2p – 3q.
Answer 3
2p – 3q = 2(2i – 8j) – 3(-9i + j)
= 4i – 16j + 27i – 3j
= 31i – 19j
Elementary Vector Algebra in Two Dimensions
Example 4
Forces F1 = (6i + 4j)N, F2 = (-2i – 5j)N and F3 = (ai + bj)N are in
equilibrium. Find the value of a and b.
Answer 4
If F1, F2 and F3 are in equilibrium then:
F1 + F2 + F3 = 0
so
(6i + 4j) + (-2i – 5j) + (ai + bj) = 0
(6 – 2 + a)i + (4 – 5 + b)j = 0
(4 + a)i + (-1 + b)j = 0
So the i components and the j components must be zero
4+a=0
-1 + b = 0
=>
a = -4
=>
b=1
Using Vectors To Describe Motion in the Plane
In Topic 1 we studied the constant acceleration formulae for
motion in one dimension:

v = u + at

s = ut + ½at2

s = ½(u + v)t
Using vectors we can now use similar formulae to study motion
in two dimensions:

v = u + at

s = ut + ½at2

s = ½(u + v)t
Using Vectors To Describe Motion in the Plane
In Topic 3 we studied Newton’s Second Law for motion in one
dimension:

F = ma
Using vectors we can now use a similar equation to study motion
in two dimensions:

F = ma
Example 5
Initially a particle P of mass 3kg is at rest at the point D whose
displacement relative to the origin O is OD = (i + 2j)metres. If
the force F = (9i + 3j)newtons acts on P, find the displacement
OE of P from O after 2 seconds.
Using Vectors To Describe Motion in the Plane
Example 6
A particle of mass 5kg is acted upon by a constant force of:
(15i + 20j)newtons
If, after 8 seconds, its velocity is:
(24i + 20j)m/s
Find its initial velocity.