Chapter 3
Motion in Two Dimensions
Units of Chapter 3
• Components of Motion
• Vector Addition and Subtraction
• Projectile Motion
• Relative Velocity
3.1 Components of Motion
An object in motion on a plane can be located
using two numbers—the x and y coordinates of its
position. Similarly, its velocity can be described
using components along the x- and y-axes.
3.1 Components of Motion
The velocity components are:
The magnitude of the velocity vector is:
3.1 Components of Motion
The components of the displacement are then
given by:
Note that the x- and y-components are
calculated separately.
3.1 Components of Motion
The equations of motion are:
When solving two-dimensional kinematics
problems, each component is treated
separately. The time is common to both.
3.1 Components of Motion
If the acceleration is not
parallel to the velocity,
the object will move in a
curve:
3.2 Vector Addition and Subtraction
Geometric methods of vector addition
Triangle method:
3.2 Vector Addition and Subtraction
The negative of a vector has the same
magnitude but is opposite in direction to the
original vector. Adding a negative vector is the
same as subtracting a vector.
3.2 Vector Addition and Subtraction
Vector Components and the Analytical
Component Method
If you know A and B,
here is how to find C:
3.2 Vector Addition and Subtraction
The components
of C are given by:
Equivalently,
3.2 Vector Addition and Subtraction
Vectors can also be written using unit vectors:
3.2 Vector Addition and Subtraction
Vectors can be resolved into components and
the components added separately; then
recombine to find the resultant.
3.2 Vector Addition and Subtraction
This is done most easily if all vectors start at the
origin.
3.3 Projectile Motion
An object projected
horizontally has an
initial velocity in the
horizontal direction,
and acceleration (due
to gravity) in the
vertical direction. The
time it takes to reach
the ground is the same
as if it were simply
dropped.
3.3 Projectile Motion
A projectile launched in an arbitrary direction
may have initial velocity components in both
the horizontal and vertical directions, but its
acceleration is still downward.
3.3 Projectile Motion
The vertical motion is the same as if the object
were thrown straight up or down with the same
initial y velocity, and the horizontal velocity is
constant.
3.3 Projectile Motion
The range of a projectile is maximum (if there is no
air resistance) for a launch angle of 45°.
3.3 Projectile Motion
With air resistance, the range is shortened,
and the maximum range occurs at an angle
less than 45°.
3.4 Relative Velocity
Velocity may be
measured in any
inertial reference
frame. At top, the
velocities are
measured relative to
the ground; at
bottom they are
measured relative to
the white car.
3.4 Relative Velocity
In two dimensions, the components of the velocity,
and therefore the angle it makes with a coordinate
axis, will change depending on the point of view.
Review of Chapter 3
• Two-dimensional motion is analyzed by
considering each component separately. Time is
the common factor.
Review of Chapter 3
• Vector components:
• In projectile motion, the horizontal and
vertical motions are determined
separately.
Review of Chapter 3
• Range is the maximum horizontal distance
traveled.
• Relative velocity is expressed relative to a
particular reference frame.