Chapter 3
Kinematics in Two Dimensions
3.1 Displacement, Velocity, and Acceleration
ro initial position
r final position
r r ro displaceme nt
3.1 Displacement, Velocity, and Acceleration
Average velocity is the
displacement divided by
the elapsed time.
r ro r
v
t to t
3.1 Displacement, Velocity, and Acceleration
The instantaneous velocity indicates how fast
the car moves and the direction of motion at each
instant of time.
r
v lim
t 0 t
3.1 Displacement, Velocity, and Acceleration
r
v lim
t 0 t
3.1 Displacement, Velocity, and Acceleration
DEFINITION OF AVERAGE ACCELERATION
v v o v
a
t to t
v
v
vo
3.2 Equations of Kinematics in Two Dimensions
Equations of Kinematics
v vo at
x 1
2
vo v t
2 2
v v 2axo
2
x vot at 1
2
3.2 Equations of Kinematics in Two Dimensions
v x vox a x t x 1
2
vox vx t
2 2
x voxt a x t 1 2 v v 2a x x
x ox
2
3.2 Equations of Kinematics in Two Dimensions
v y voy a y t
2
y voyt a y t 1
2
y 1
2
v oy vy t
2 2
v v 2a y y
y oy
3.2 Projectile Motion Diagram
The x part of the motion occurs exactly as it would if the
y part did not occur at all, and vice versa.
3.2 Projectile Motion Diagram
The x part of the motion occurs exactly as it would if the
y part did not occur at all, and vice versa.
3.2 Equations of Kinematics in Two Dimensions
Example 1 A Moving Spacecraft
In the x direction, the spacecraft has an initial velocity component
of +22 m/s and an acceleration of +24 m/s2. In the y direction, the
analogous quantities are +14 m/s and an acceleration of +12 m/s2.
At a time 7.0 s, find (a) x and vx, (b) y and vy, and (c) the final
velocity of the spacecraft.
3.2 Equations of Kinematics in Two Dimensions
Example 1 A Moving Spacecraft
In the x direction, the spacecraft has an initial velocity component
of +22 m/s and an acceleration of +24 m/s2. In the y direction, the
analogous quantities are +14 m/s and an acceleration of +12 m/s2.
Find (a) x and vx, (b) y and vy, and (c) the final velocity of the
spacecraft at time 7.0 s.
x ax vx vox t
? +24.0 m/s2 ? +22 m/s 7.0 s
y ay vy voy t
? +12.0 m/s2 ? +14 m/s 7.0 s
3.2 Equations of Kinematics in Two Dimensions
x ax vx vox t
? +24.0 m/s2 ? +22 m/s 7.0 s
2
x voxt a x t
1
2
22 m s 7.0 s 1
2 24 m s 7.0 s
2 2
740 m
v x vox a x t
22 m s 24 m s 2
7.0 s 190 m s
3.2 Equations of Kinematics in Two Dimensions
y ay vy voy t
? +12.0 m/s2 ? +14 m/s 7.0 s
2
y voyt a y t
1
2
14 m s 7.0 s 12 m s 1
2
2
7.0 s 2
390 m
v y voy a y t
14 m s 12 m s 2
7.0 s 98 m s
3.2 Equations of Kinematics in Two Dimensions
v
v y 98 m s
v x 190 m s
v 190 m s 2
98 m s 210 m s
2
tan 1
98 190 27
3.2 Equations of Kinematics in Two Dimensions
3.3 Projectile Motion
Under the influence of gravity alone, an object near the
surface of the Earth will accelerate downwards at 9.80m/s2.
a y 9.80 m s 2
ax 0
v x vox constant
3.3 Projectile Motion
Example 2 A Falling Care Package
The airplane is moving horizontally with a constant velocity of
+115 m/s at an altitude of 1050m. Determine the time required
for the care package to hit the ground.
3.3 Projectile Motion
y ay vy voy t
-1050 m -9.80 m/s2 0 m/s ?
3.3 Projectile Motion
y ay vy voy t
-1050 m -9.80 m/s2 0 m/s ?
2 2
y voyt a y t 1
2 y a yt
1
2
2y 2 1050 m
t 2
14 .6 s
ay 9.80 m s
3.3 Projectile Motion
Example 3 The Velocity of the Care Package
What are the magnitude and direction of the final velocity of
the care package?
3.3 Projectile Motion
y ay vy voy t
-1050 m -9.80 m/s2 ? 0 m/s 14.6 s
3.3 Projectile Motion
y ay vy voy t
-1050 m -9.80 m/s2 ? 0 m/s 14.6 s
v y voy a y t 0 9.80 m s 14.6 s 2
143 m s
v vx vy 2 2
3.3 Projectile Motion
Conceptual Example 4 I Shot a Bullet into the Air...
Suppose you are driving a convertible with the top down.
The car is moving to the right at constant velocity. You point
a rifle straight up into the air and fire it. In the absence of air
resistance, where would the bullet land – behind you, ahead
of you, or in the barrel of the rifle?
3.3 Projectile Motion
Example 5 The Height of a Kickoff
A placekicker kicks a football at and angle of 40.0 degrees and
the initial speed of the ball is 22 m/s. Ignoring air resistance,
determine the maximum height that the ball attains.
3.3 Projectile Motion
vo
voy
vox
voy vo sin 22 m s sin 40 14 m s
vox vo sin 22 m s cos 40 17 m s
3.3 Projectile Motion
y ay vy voy t
? -9.80 m/s2 0 14 m/s
3.3 Projectile Motion
y ay vy voy t
? -9.80 m/s2 0 14 m/s
2 2
2 2
v v 2a y y v v
y oy
y oy y
2a y
0 14 m s
2
y 10 m
2 9.8 m s
2
3.3 Projectile Motion
Example 6 The Time of Flight of a Kickoff
What is the time of flight between kickoff and landing?
3.3 Projectile Motion
y ay vy voy t
0 -9.80 m/s2 14 m/s ?
3.3 Projectile Motion
y ay vy voy t
0 -9.80 m/s2 14 m/s ?
2
y voyt a y t
1
2
0 14 m s t 1
2 9.80 m s t2 2
0 214 m s 9.80 m s t 2
t 0, t 2.9 s
3.3 Projectile Motion
Example 7 The Range of a Kickoff
Calculate the range R of the projectile.
2
x voxt a x t voxt
1
2
17 m s 2.9 s 49 m
3.3 Projectile Motion
Conceptual Example 9 Two Ways to Throw a Stone
From the top of a cliff, a person throws two stones. The stones
have identical initial speeds, but stone 1 is thrown downward
at some angle above the horizontal and stone 2 is thrown at
the same angle below the horizontal. Neglecting air resistance,
which stone, if either, strikes the water with greater velocity?
3.4 Relative Velocity
v PG v PT v TG
3.4 Relative Velocity
Example 10 Crossing a River
The engine of a boat drives it across a river that is 1800m wide.
The velocity of the boat relative to the water is 4.0m/s directed
perpendicular to the current. The velocity of the water relative
to the shore is 2.0m/s.
(a) What is the velocity of the
boat relative to the shore?
(b) How long does it take for
the boat to cross the river?
3.4 Relative Velocity
v BS v BW v WS
4.0
1
tan 63
2.0
vBS v 2
BW
2
v
WS 4.0 m s 2.0 m s
2 2
4.5 m s
3.4 Relative Velocity
1800 m
t 450 s
4.0 m s
Problem Solving:
1. A spacecraft is traveling with a velocity of v0x = 5480 m/s
along the +x direction. Two engines are turned on for a
time of 842 s. One engine gives the spacecraft an
acceleration in the +x direction of ax = 1.20 m/s2, while the
other gives it an acceleration in the +ydirection of ay = 8.40
m/s2. At the end of the firing, find (a) vx and (b) vy
2. As a tennis ball is struck, it departs from the racket
horizontally with a speed of 28.0 m/s. The ball hits the
court at a horizontal distance of 19.6 m from the racket.
How far above the court is the tennis ball when it leaves
the racket?
Problem Solving:
3. A batter hits a baseball so that it leaves the bat at speed at
an angle 53.1°. (a) Find the position of the ball and its
velocity (magnitude and direction) at (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.
4. You throw a ball from your window 8.0 m above the
ground. When the ball leaves your hand, it is moving at at
an angle of below the horizontal. How far horizontally
from your window will the ball hit the ground? Ignore air
resistance.
5. A stone is thrown from the top of a building upward at an
angle of 30.0° to the horizontal with an initial speed of 20.0
m/s, as shown in the Fig. If the height of the building is
45.0 m,
a) how long does it take the stone to reach the ground?
b) What is the speed of the stone just before it strikes the
ground?
