1. business and industrial
2. shipbuilding

Physics 218
Lecture 5: Kinematics
in 2-3D
Alexei Safonov
Checklist for Today
•  For the rest of this week:
–  Be ready for the lab (WebAssign) including pre-labs
•  Did you get my email about the question on the uncertainty for volume in
problem #4?
–  Quiz at the recitation on Chapter 2
–  There was a pre-lecture for today
•  Many people completed, but not all
–  A pre-lecture due Wednesday morning
•  Don’t forget the checkpoints
–  Homework for Chapter 2 is due Sunday on MP
•  Coming week recitation quiz will be on Chapter 3
Speeder
A speeder passes you (a police officer) sitting by the side of
the road and maintains their constant velocity V. You
immediately start to move after the speeder from rest with
constant acceleration a.
• How much time does it take to ram the speeder?
• How far do you have to travel to catch the speeder?
• What is your final speed?
Police Officer
Speeder
X
Throw a Ball up
You throw a ball upward into the air with
initial velocity V0. Calculate:
a)  The time it takes to reach its highest
point (the top).
b)  Distance from your hand to the top
c)  Time to go from your hand and come
back to your hand
d)  Velocity when it reaches your hand
e)  Time from leaving your hand to reach
some random height h.
Chapter 3
•  Kinematics in Two or Three Dimensions
•  Projectile Motion
•  Uniform Circular Motion
Important Equations of Motion
If the acceleration is constant

 
v = v 0 + at

 

2
1
x = x 0 + v 0 t + 2 at
Position, velocity and Acceleration are
vectors.
Projectile Motion
The horizontal and vertical
equations of the motion behave
independently
Problem solving:
The trick for all these problems is to break
them up into the X and Y directions.
Constant Acceleration
x = x 0 +v 0x t + a x t
1
2
2
  
2
2
1
1
R = r0 +v 0 t + 2 at y = y0 +v 0y t + 2 a y t
  
v = v 0 + at
v x = v 0x +a x t
v y = v 0y +a y t
Kinematics in 3D
Projectile Motion & Frames of
Reference
Checkpoint 1
A physics demo launches one marble horizontally while at the same instant dropping a second marble straight down. Which one hits the ground first? A) The launched marble hits first. B) The dropped marble hits first. C) They both hit at the same :me. Lets check again Ball Dropping
•  Analyze Vertical
and Horizontal
separately!!!
•  Ay = g (downwards)
•  Ax = 0
–  Constant for Both
cases!!!
Vx = 0
Vx>0
Projectile Motion
The horizontal and vertical
equations of the motion behave
independently
Problem solving:
The trick for all these problems is to break
them up into the X and Y directions.
Projectile Motion
Horizontal
Vertical
Boring
Monkey Troubles
• You are a vet trying to shoot a tranquilizer dart into a monkey
hanging from a branch in a distant tree. You know that the
monkey is very nervous, and will let go of the branch and start
to fall as soon as your gun goes off. In order to hit the monkey
with the dart, where should you point the gun before shooting?
• A) Right at the monkey
• B) Below the monkey
• C) Above the monkey
Shooting the Monkey…
Dart x = vo t
1
y = − gt 2
2
Monkey
x = xo
1
y = − gt 2
2
Shooting the Monkey…
S:ll works even if you shoot upwards! y = voy t - 1/2 g t 2
y = yo - 1/2 g t 2
Dart hits the monkey • 
Checkpoint
2
A destroyer simultaneously fires two shells with the same initial
speed at two different enemy ships. The shells follow the
trajectories shown. Which ship gets hit first.
Destroyer Enemy 1 A) Enemy 1 B) Enemy 2 C) They are both hit at the same :me Enemy 2 Checkpoint 2
•  …Which enemy ship gets hit first?
•  A) Enemy 1 B) Enemy 2 C) Same
Destroyer Enemy 1 Enemy 2 A) they are traveling at the same speed, but the enemy one trajectory is shorter B) Both shots were accelera:ng towards the ground at the same rate, but the shot fired at Enemy 2 did not go as high and therefore took less :me to fall back to the ground. C) we are given that two shells are fired at same speed. therefore, both ships should get hit at the same :me. Checkpoint 3
• 
A destroyer fires two shells with different initial speeds at two
different enemy ships. The shells follow the trajectories
shown. Which enemy ship gets hit first?
Destroyer Enemy 1 A) Enemy 1
B) Enemy 2
C) They are both hit at the same time
Enemy 2 Checkpoint 3
•  …Which enemy ship gets hit first?
•  A) Enemy 1 B) Enemy 2 C) Same
Destroyer Enemy 1 Enemy 2 A) Since the ini:al speed for the first shell is lower, Enemy Ship 1 will be hit first. B) The ini:al speed of the shell fired at ship 2 is greater, so ship 2 gets hit first. C) they both achieve the same height so they remain in the air the same amount of :me Marbles & Math
•  Prove
mathematically
that an object
projected
horizontally will
reach the ground
at the same time
as an object
dropped vertically
FINISHED HERE
Firing up in the air at an angle
A ball is fired up in the air with velocity Vo and
angle Θo. Ignore air friction. The acceleration
due to gravity is g pointing down.
What is the final velocity here?
Football Punt
•  A football is kicked at angle Θ0 with a
velocity V0. The ball leaves the punters foot
h meters above the ground.
–  The velocity at the maximum height
–  How far does it travel, in the X direction, before
it hits the ground?
–  What angle maximizes the distance traveled
h
In the previous problem, which of
the given angles minimizes the
horizontal distance traveled?
A. 
B. 
C. 
D. 
θ=10 degrees
θ=30 degrees
θ=60 degrees
θ=90 degrees
Checklist for Today
•  For the rest of this week:
–  Be ready for the lab (WebAssign) including pre-labs
–  Quiz at the recitation on Chapter 3
–  There was a pre-lecture for today
•  Many people completed, but not all
–  A pre-lecture due Wednesday morning
•  Don’t forget the checkpoints
–  Homework for Chapter 3 is due Sunday on MP
•  Coming week recitation quiz will be on Chapter 3
Uniform Circular Motion
•  Fancy words for moving in a circle with
constant speed
•  We see this around us all the time
–  Moon around the earth
–  Earth around the sun
–  Merry-go-rounds
Uniform Circular Motion Velocity
•  Velocity vector = |
V| tangent to the
circle
•  Is this ball
accelerating?
– Why?
Centripetal Acceleration


 
a = dv / dt ≈ (v2 − v1 ) / dt
R
a
•  Vector difference V2 - V1 gives the
direction of acceleration a
Centripetal Acceleration
•  “Center Seeking”

•  Accel vector= V2/R a =
towards the center
•  Acceleration is
perpendicular to
rˆ direction
velocity
2
v
(−rˆ)
R
R
Circular Motion: Get the speed!
Speed = distance/time
! Distance
in 1 revolution divided by
the time it takes to go around once
!Speed
= 2πr/T
Note: The time to go around once is
known as the Period, or T
Ball on a String
•  A ball at the end of a string is
revolving uniformly in a horizontal
circle (ignore gravity) of radius R.
The ball makes N revolutions in a
time t.
•  What is the centripetal
acceleration?
In previous problem, how would
acceleration change if the number
of revolutions N were to double?
A.  Acceleration will be ½ of the original
value
B.  Acceleration with be double original
value
C.  Acceleration will be quadruple original
value
D.  Acceleration will be ¼ of the original
value
Firing up in the air at an angle
A ball is fired up in the air with velocity Vo and
angle Θo. Ignore air friction. The acceleration
due to gravity is g pointing down.
What is the final velocity here?
Boat on the River
•  You want to cross the
river so that the boat
gets exactly from A to
B. The river has a
current vC=4 km/h.
Your boat’s speed in
still water is
vB=20km/h?
•  What is the angle θ
you should aim at to
do that?
θ
vB
In previous problem, is it possible
to get from A to B for any values for
vB and vC?
A. 
B. 
C. 
D. 
Yes, always possible
Only possible if vB>vC
Only possible if vB>2vC
Only possible if vB>>vC (much larger)
vC
θ
vB
Next time…
•  Reading: Finish Chapter 3 if you haven’t
already
•  Homework:
–  HW1 was due yesterday
–  HW2 covered in recitation this week; due
Monday (6 days from now)
–  Start working on HW3
•  Next time: More on kinematics in two
dimensions
–  Reading Quiz hints: Q3.13-Q3.16
A pendulum swings in an arc, at what
point (of the three points A, B, C) is the
magnitude of the acceleration in the x
direction greatest?
A. 
B. 
C. 
D. 
A
B
C
Same at all points
After leaving the gun a projectile moves
in a parabolic path without air
resistance. Which statement is TRUE?
A.  a is parallel to v along its path.
B.  a is perpendicular to v at top of the
parabola.
C.  a is perpendicular to v along its path.
D.  a is parallel to v at the top of the
parabola.