Physics for Dentistry and
Medicine students
PHYS 145
Text book
Physics; John D. Cutnell and Kenneth W.
Johnson; 7th edition; Wiley; 2007.
Chapter 3
Kinematics in Two
Dimensions
Once this mountain lion is airborne, it follows a path that depends on the launch
velocity and the acceleration due to gravity, assuming that the effects of air
resistance can be ignored. (George H. H. Huey/Corbis Images)
Displacement, Velocity, and
Acceleration
• Definition: The
displacement is the vector
that points from an object’s
initial position to its final
position and has a
magnitude that equals the
shortest distance between
the two positions.
• SI Unit of displacement:
meter (m)
  
Displaceme nt   r  r  ro
Average Velocity
• Definition of Average Velocity
Displaceme nt
Average velocity 
Elapsed time
 

 r  ro  r
v 

t  to
t
• SI Unit of Average Velocity and Speed:
meter per second (m/s)
Instantaneous Velocity
• The instantaneous velocity v of the car
indicates how fast the car moves and the
direction of the motion at each instant of
time. The magnitude of the instantaneous
velocity is called the instantaneous
velocity, and it is the number (with units)
indicated by the speedometer


r
v  lim
t  0  t
Acceleration
• Definition of Average Acceleration
Change in velocity
Average acceleration 
Elapsed time
 

 v  vo  v
a

t  to
t
SI Unit of Average Acceleration: (m/s2)
• The instantaneous acceleration


v
a  lim
t 0  t
Equations of Kinematics in
two Dimensions
x Component
x
ax
vx
vox
t
vx = vox + ax t
x = ½ (vox + vx) t
x = vox t + ½ ax t2
vx2 = vox2 + 2ax x
Variable
Displacement
Acceleration
Final velocity
Initial velocity
Elapsed time
y Component
y
ay
vy
voy
t
vy = voy + ay t
y = ½ (voy+vy) t
y = voy t + ½ ay t2
vy2 = voy2 + 2ay y
Projectile Motion
Important notes:
1. The x component of the velocity remains
constant at its initial value or vx = vox.
2. The x component of the acceleration is ax = 0
m/s2.
3. The ay is the downward acceleration due to
gravity
ay = g = 9.80 m/s2 or 32.2 ft/s2 .
4. The time to reach maximum height from any
point = the time spent returning from the
maximum height to that point.
5. vy = 0 m/s at maximum height.
Example
An airplane moving
horizontally with a constant
velocity of +115 m/s at an
altitude of 1050 m. The
directions to the right and
upward have been chosen as
the positive directions. The
plane releases a “care
package” that falls to the
ground along a curved
trajectory. Ignoring air
resistance, determine the time
required for the package to hit
the ground.
Solution
y - Direction Data
y
ay
vy
- 1050 m
- 9.80 m/s2
voy
T
0 m/s
?
Substitute in the following equation of
motion, the falling time can be calculated.
y = voy t + ½ ay t2
where t = 14.6 s
Example 6: The Height of a
Kickoff
A placekicker kicks a football at an angle of
θ = 40.0 o above the horizontal axis, as in
the figure. The initial speed of the ball is vo
= 22 m/s. Ignore air resistance and find the
maximum height H that the ball attains.
Solution
To calculate the vertical component
voy = vo sin θ = + (22 m/s) sin 40.0o = + 14 m/s
y - Direction Data
y
ay
vy
voy
H=?
- 9.80 m/s2
0 m/s
+14 m/s
vy2 = voy2 + 2ay y
y = H = + 10 m
t
Solution
To determine the time of flight between
kickoff and landing
y - Direction Data
y
ay
vy
0m
- 9.80 m/s2
voy
t
+14 m/s
?
y = voy t + ½ ay t2
where t = 0 s or 2.9 s
Solve This Problem
• 1/307) A jetliner is moving at a speed of
245 m/s. The vertical component of the
plane’s velocity is 40.6 m/s. Determine
the magnitude of the horizontal
component of the plane’s velocity.
H.W.
3/307) A mountain-climbing expedition establishes
two intermediate camps, labeled A and B in the
drawing, above the base camp. What is the
magnitude r of the displacement between camp A
and camp B?
Problem
12/309) On a spacecraft, two engines are
turned on for 684 s at a moment when the
velocity of the craft has x and y components of
vox = 4370 m/s and voy = 6280 m/s. While the
engines are firing, the craft undergoes a
displacement that has components of x = 4.11 x
106 m and y = 6.07 x 106 m. Find the x and y
components of the craft’s acceleration.
Problem
14/309) A volleyball is spiked so that it has an
initial velocity of 15 m/s directed downward at
an angle of 55° below the horizontal. What is the
horizontal component of the ball’s velocity when
the opposing player fields the ball?
Quiz
Two trees have perfectly straight trunks and are
both growing perpendicular to the flat horizontal
ground beneath them. The sides of the trunks that face
each other are separated by 1.3 m. A frisky squirrel
makes three jumps in rapid succession. First, he leaps
from the foot of one tree to a spot that is 1.0 m above
the ground on the other tree. Then, he jumps back to
the first tree, landing on it at a spot that is 1.7 m above
the ground. Finally, he leaps back to the other tree,
now landing at a spot that is 2.5 m above the ground.
What is the magnitude of the squirrel’s displacement?