Physics for Engineers
Instructor: Engr. Aquim P. Bordomeo, CIT University
3 Motion in Two or Three Dimensions
Kinematic Equations for Two-
Dimensional Motion
⚫ When the two-dimensional motion has a constant
acceleration, a series of equations can be developed that
describe the motion
⚫ These equations will be similar to those of one-
dimensional kinematics
⚫ Motion in two dimensions can be modeled as two
independent motions in each of the two perpendicular
directions associated with the x and y axes
⚫ Any influence in the y direction does not affect the motion in the
x direction
Kinematic Equations, 2
⚫ Position vector for a particle moving in the xy plane
⚫ The velocity vector can be found from the position
vector r = x ˆi + yˆj
dr
v= = v x ˆi + v y ˆj
dt
⚫ Since acceleration is constant, we can also find an
expression for the velocity as a function of time:
v f = v i + at
Projectile Motion
⚫ An object may move in both the x and y
directions simultaneously
⚫ The form of two-dimensional motion we will
deal with is called projectile motion
Assumptions of Projectile Motion
⚫ The free-fall acceleration is constant
over the range of motion.
⚫ It is directed downward
⚫ This is the same as assuming a flat Earth
over the range of the motion
⚫ It is reasonable as long as the range is
small compared to the radius of the Earth
⚫ The effect of air friction is negligible.
⚫ With these assumptions, an object in
projectile motion will follow a parabolic
path
⚫ This path is called the trajectory.
Projectile Motion Diagram
ax = 0
The x-direction has constant velocity.
So, ax = 0.
The y-direction is free fall.
The initial velocity can be expressed in
terms of its components.
.
v xi = v i cos
v yi = v i sin
Two Types
⚫ Horizontal Launch (kicking a stone off a
bridge)
⚫ Angled Launch (golf, baseball)
- the motion is uniquely defined by its
a. launch angle ()
b. initial velocity (vi)
Range and Maximum Height of a Projectile
When analyzing projectile
motion, two characteristics
are of special interest
1. The range, R, is the
horizontal distance of
the projectile.
2. The maximum height
the projectile reaches
is h.
Height of a Projectile, equation
⚫ Themaximum height of the projectile can be
found in terms of the initial velocity vector:
v sin i
2 2
h= i
2g
⚫ This
equation is valid only for symmetric
motion
Range of a Projectile, equation
⚫ The range of a projectile can be expressed in terms
of the initial velocity vector:
v sin2 i
2
R= i
g
⚫ This is valid only for symmetric trajectory
Time to Maximum Height, equation
v i sin i
t=
g
Total Time in Air, equation
2 v i sin i
ttotal =
g
More About the Range of a Projectile
Range of a Projectile, final
⚫ The maximum range occurs at i = 45o
⚫ Complementary angles will produce the same
range
⚫ The maximum height will be different for the two angles
⚫ The times of the flight will be different for the two angles
Projectile Motion – Problem Solving Hints
⚫ Conceptualize
⚫ Establish the mental representation of the projectile moving
along its trajectory
⚫ Categorize
⚫ Confirm air resistance is neglected
⚫ Select a coordinate system with x in the horizontal and y in the
vertical direction
⚫ Analyze
⚫ If the initial velocity is given, resolve it into x and y components
⚫ Treat the horizontal and vertical motions independently
Projectile Motion – Problem Solving
Hints, cont.
⚫ Analysis, cont
⚫ Analyze the horizontal motion using constant velocity
techniques
⚫ Analyze the vertical motion using constant acceleration
techniques
⚫ Remember that both directions share the same time
⚫ Finalize
⚫ Check to see if your answers are consistent with the mental and
pictorial representations
⚫ Check to see if your results are realistic
Non-Symmetric Projectile Motion
⚫ Follow the general rules for
projectile motion
⚫ Break the y-direction into
parts
⚫ up and down or
⚫ symmetrical back to initial
height and then the rest of
the height
⚫ Apply the problem solving
process to determine and
solve the necessary
equations
⚫ May be non-symmetric in
other ways
SAMPLE PROBLEMS
1. A motorcycle stunt rider rides off the edge of
a cliff. Just at the edge his velocity is
horizontal, with magnitude 9 m/s. Find the
motorcycle’s
a.distance from the edge of the cliff, and
b.velocity
0.50 s after it leaves the edge of the cliff.
SAMPLE PROBLEMS
At any time t,
x = v xi t
1 2
y = v yi t + gt
2
( x ) + ( y ) distance of the projectile
2 2
r= from the origin
At any time t,
v x = v xi Magnitude of Projectile’s velocity
v = vx + vy
2 2
v y = v yi + gt
Direction of Projectile’s velocity
v
= tan −1 y
vx
SAMPLE PROBLEMS
2. A batter hits a baseball so that it leaves the bat at
speed of 37 m/s at an angle 53.1º .
(a) Find the position of the ball and its velocity
(magnitude and direction) at t = 2 seconds
(b) Find the time when the ball reaches the
highest point of its flight, and its height h at
this time.
(c) Find the horizontal range R—that is, the
horizontal distance from the starting point to
where the ball hits the ground.
SAMPLE PROBLEMS
3. A stone is thrown upward from the top
of a building at an angle of 30.0° to the
horizontal and with an initial speed of
20.0 m/s, as in Figure 3.19. The point
of release is 45.0 m above the ground.
(a) How long does it take for the stone to
hit the ground?
(b) Find the stone’s speed at impact.
(c) Find the horizontal range of the stone.
Neglect air resistance.
SAMPLE PROBLEMS
ACTIVITY
1. A physics book slides off a horizontal tabletop with a
speed of 1.1 m/s. It strikes the floor in 0.350 s. Ignore air
resistance. Find
(a) the height of the tabletop above the floor;
(b) the horizontal distance from the edge of the table
to the point where the book strikes the floor;
(c) the horizontal and vertical components of the
book’s velocity, and the magnitude and direction of
its velocity, just before the book reaches the floor
ACTIVITY
2. A rookie quarterback throws a football with an initial
upward velocity component of 12 m/s and a horizontal
velocity component of 20 m/s. Ignore air resistance.
(a) What is the initial velocity?
(b) How much time is required for the football to reach the
highest point of the trajectory? How high is this point?
(c) How much time (after it is thrown) is required for the
football to return to its original level?
(d) What is the position of the football at t=1 s?
(e) What is the velocity of the football at t=1 s?
A man stands on the roof of a 15.0-m-tall building and
throws a rock with a velocity of magnitude 30 m/s at an
angle of 33° above the horizontal. You can ignore air
resistance. Calculate
(a) the maximum height above the roof reached by the
rock;
(b) the magnitude of the velocity of the rock just before
it strikes the ground; and
(c) the horizontal range from the base of the building
to the point where the rock strikes the ground.
A ball is thrown at 20 m/s from the roof
of a building 25 high at an angle of 30
degrees below the horizontal. At what height
above the ground will the ball strike the side
of another building 20 meters away from the
first?
