Chapter 3
Motion in Two and Three
Dimensions; Vectors
Vectors and Scalars
A vector has magnitude as
well as direction.
Some vector quantities:
displacement, velocity,
force, momentum
A scalar has only a
magnitude.
Some scalar quantities:
mass, time, temperature
Addition of Vectors – Graphical Methods
Adding the 2 vectors
Subtracting 2 vectors
Addition of Vectors – Graphical Methods
Consider a motion in two dimensions. Suppose, you move
to the right 8.0 km and then 4.0 km up. What is your
displacement?
By using the Pythagorean
Theorem, we have
DR  (8km) 2  (4km) 2
 8.9 km
Adding Vectors by Components
Any vector can be expressed as the sum
of two other vectors, which are called its
components. Usually the other vectors are
chosen so that they are perpendicular to
each other.
Adding Vectors by Components
The components are found
using trigonometric functions.
Vx  V cos 
V y  V sin 
tan  
Vy
Vx
V  Vx  V y
2
2
2
Adding Vectors by Components
Adding vectors:
1. Draw a diagram
2. Choose x and y axes.
3. Resolve each vector into x and y components.
4. Calculate each component using sines and cosines.
5. Add the components in each direction.
6. To find the length and direction of the vector, use:
Adding Vectors by Components
Vx  V1 x  V2 x
Components:
Resultant:
Direction:
V y  V1 y  V2 y
Vector V1 is 6.6 units long and points along the negative x axis.
Vector V2 is 8.5 units long and points at an angle of 45o to the
positive x axis. (a) What are the x and y components of each
vector? (b) Determine the sum (magnitude and angle).
8.(a)
V1x  6.6 units
V1 y  0 units
V2 x  8.5cos 45o  6.0 units
V2 y  8.5sin 45o  6.0 units
Vx  V1x  V2 x  6.6  6.0  0.6 units
(b)
Vy  V1 y  V2 y  0  6.0  6.0 units
V=
V1 + V2 
 0.6 2   6.0 2
 6.0 units
  tan 1
6.0
0.6
 84 o
The sum has a magnitude of 6.0 units, and is 84o clockwise from the –
negative x-axis, or 96o counterclockwise from the positive x-axis.
Unit vectors i and j
Express each vector as the sum of 2 perpendicular vectors. It is common to
use the horizontal and vertical directions using unit vectors i and j
Example:
A = Ax + Ay= Axi + Ayj , where Ax and Ay are the horizontal and vertical
components
B = Bx + By = Bxi + Byj, where Bx and By are the horizontal and vertical
components
Some useful properties of unit vectors
i.j=0; i.i=1; j.j=1
A.B = AxBx+AyBy
Three vectors are expressed as A = 4i – j, B = -3i + 2j, and
C = -3j. If R = A+ B + C, find the magnitude and direction of R.
R = A+ B + C = i-2j
magnitude = 2.24
angle = 63.4 o, below the x-axis
Projectile Motion
A projectile is an object
moving in two
dimensions under the
influence of Earth's
gravity; its path is a
parabola.
Projectile Motion
This photograph shows two balls
that start to fall at the same time.
The one on the right has an initial
speed in the x-direction. It can be
seen that vertical positions of the
two balls are identical at identical
times, while the horizontal
position of the yellow ball
increases linearly.
Projectile Motion
A projectile can be understood
by analyzing the horizontal and
vertical motions separately.
The speed in the x-direction is
constant
in the y-direction the object
moves with constant
acceleration g.
Projectile Motion
If an object is launched at an initial angle of θ0
with the horizontal, the analysis is similar except
that the initial velocity has a vertical component.
Solving Problems Involving Projectile
Motion
Projectile motion is motion with constant
acceleration in two dimensions, where the
acceleration is g and is down.
A diver running 1.8 m/s dives out horizontally from the edge of a vertical cliff
and 3.0 s later reaches the water below. How high was the cliff, and how far
from its base did the diver hit the water?
Choose downward to be the positive y direction. The origin will be at the point where the diver
dives from the cliff. In the horizontal direction,
vx 0  1.8 m s
In the vertical direction,
ax  0
and
vy 0  0
ay  9.80 m s 2
y0  0
and the time of flight is t = 3.0 s
The height of the cliff is found from applying to the vertical motion.
y  y0  v y 0t  12 a y t 2

y  00
1
2

9.80 m s 2
  3.0 s 
2
 44 m
The distance from the base of the cliff to where the diver hits the water is found from
the horizontal motion at constant velocity:
x  vxt  1.8m s  3 s   5.4 m
A football is kicked at ground level with a speed of 18.0 m/s at an angle of
35.0º to the horizontal. How much long later does it hit the ground?
Choose the point at which the football is kicked the origin, and choose upward to be the positive
y direction. When the football reaches the ground again, the y displacement is 0. For the
football,


v y 0  18.0 sin 35.0o m s Vy0
ay  9.80 m s 2
V0
35.0 o
Vx0
Vx0
- Vy0
and the final y velocity will be the opposite of the starting y velocity. to find the time of flight use
v y  v y 0  at

t
vy  vy 0
a
18.0sin 35.0  m s  18.0sin 35.0  m s



o
o
9.80 m s
2
2.11 s
Extra Slides
A fire hose held near the ground shoots water at a speed of 6.8 m/s. At what
angle(s) should the nozzle point in order that the water land 2.0 m away ? Why
are there two different angles? Sketch the two trajectories.
19.
Apply the range formula
R
v02 sin 2 0
g
sin 2 0 
Rg
v02

 2.0 m   9.8 m

 6.8 m s 2
2 0  sin 1 0.4239

s2
  0.4239
 0  13o , 77 o
There are two angles because each angle gives the same range. If one angle is
2.5
  45  
o
, then
  45o  
is also a solution. The two paths are shown in the graph.
2
1.5
1
0.5
0
0
-0.5
0.5
1
1.5
2
Addition of Vectors – Graphical Methods
Even if the vectors are not at right
angles, they can be added graphically by
using the “tail-to-tip” method.
Addition of Vectors – Graphical Methods
The parallelogram method may also be used;
Subtraction of Vectors, and
Multiplication of a Vector by a Scalar
In order to subtract vectors, we
define the negative of a vector, which
has the same magnitude but points
in the opposite direction.
Then we add the negative vector:
Subtraction of Vectors, and
Multiplication of a Vector by a Scalar
A vector V can be multiplied by a scalar c; the
result is a vector cV that has the same direction
but a magnitude cV. If c is negative, the resultant
vector points in the opposite direction.
Addition of Vectors – Graphical Methods
Adding the vectors in the opposite order gives the
same result: