Physics 201: Lecture 4

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Physics 201 Lecture 4

• Todays lecture will be on
• More on 2D motion
• Addition of velocities
• Uniform Circular Motion

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Preflight 4, Q 1 2

• Can a vector have a component bigger than its
  magnitude?
• Yes
• No

The square of magnitude of a vector is given in
terms of its components by R2 Rx 2 Ry 2 Since
the square is always positive the components
cannot be larger than the magnitude
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Preflight 4, Q 3 4

• The sum of the two components of a non-zero 2-D
  vector is zero. Which of these directions is the
  vector pointing in?
• 45o
• 90o
• 135o
• 180o

135o
-45o
The sum of components is zero implies Rx -
Ry The angle, ? tan-1(Ry / Rx) tan-1 -1
135o -45o (not unique, multiples of 2 ?)
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Summary of Lecture 3Kinematics in Two Dimensions

• x x0 v0xt 1/2 axt2
• vx v0x axt
• vx2 v0x2 2ax ?x

• y y0 v0yt 1/2 ayt2
• vy v0y ayt
• vy2 v0y2 2ay ?y

x, y and z motions are independent! They share a
common time t
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Question?
Without air resistance, an object dropped from a
plane flying at constant speed in a straight line
will 1. Quickly lag behind the plane. 2.
Remain vertically under the plane. 3. Move
ahead of the plane
There is no acceleration along horizontal -
object continues to travel at constant speed
(same as that of the plane) along horizontal. Due
to gravitational acceleration the objects speed
downwards increases.
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Lecture 5, Pre-Flight 56

• 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. On the other hand,
  you also know that the dart will not travel in a
  straight line, but rather in a parabolic path
  like any other projectile. In order to hit the
  monkey with the dart, where should you point the
  gun before shooting?
• 1 Right at the monkey
• 2 Below the monkey
• 3 Above the monkey

If the shot is fired at the monkey the same time
the monkey drops, both objects will fall at the
same rate causing the shot to hit the monkey.
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Shooting the Monkey...
x v0 t y -1/2 g t2
x x0 y -1/2 g t2
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Shooting the Monkey...
r r0 - 1/2 g t2

• At an angle, still aim at the monkey!

r v0 t - 1/2 g t2
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Shooting the Enemy Paratrooper...

• If a soldier wants to shoot down an enemy
  paratrooper descending at uniform speed, Se,
  where should he aim?
• Above the enemy
• At the enemy
• Below the enemy
• Answer depends on the enemys position and
  vertical speed.

r r0 - Se t
r v0 t - 1/2 g t2
Miss the enemy
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Problem Motion in 2D

• Two footballs are thrown from the same point on a
  flat field. Both are thrown at an angle of 30o
  above the horizontal. Ball 2 has twice the
  initial speed of ball 1. If ball 1 is caught a
  distance D1 from the thrower, how far away from
  the thrower D2 will the receiver of ball 2 be
  when he catches it? (a) D2 2D1 (b)
  D2 4D1 (c) D2 8D1

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Solution

• The distance a ball will go is simply x
  (horizontal speed) x (time in air) v0x t

• To figure out time in air, consider the
  equation for the height of the ball

• When the ball is caught, y y0

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Solution
x v0x t

• So the time spent in the air is proportional to
  v0y

• Since the angles are the same, both v0y and v0x
  for ball 2are twice those of ball 1.

• Ball 2 is in the air twice as long as ball 1, but
  it also has twice the horizontal speed, so it
  will go 4 times as far!!

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A battleship simultaneously fires two shells at
enemy ships from identical canons. If the shells
follow the parabolic trajectories shown, which
ship gets hit first? 1. Ship A. 2. Ship
B. 3. Both at the same time
Higher the shell flies, the longer it takes.
What should the captain order if he wants to hit
both ships at the same time?
A
B
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Reference Frames Relative Motion
VCBVCAVAB
Velocity of B relative to ground ( C )
VCB Velocity of A relative to ground ( C )
VCA Velocity of B relative to A VAB
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Relative Motion

• If an airplane flies in a jet stream, depending
  on the relative orientation of the airplane and
  the jet stream, the plane can go faster or slower
  than it normally would in the absence of the jet
  stream
• If a person rows a boat across a rapidly flowing
  river and tries to head directly for the shore,
  the boat moves diagonally relative to the shore
• Velocity is a vector - add velocities like
  vectors
• Sum the components
• Vx V1x V2x
• V1x V1 cosq
• Vy V1y V2y
• V1y V1 sinq

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Question

• Three swimmers can swim equally fast relative to
  the water. They have a race to see who can swim
  across a river in the least time. Relative to
  the water, Beth (B) swims perpendicular to the
  flow, Ann (A) swims upstream, and Carly (C) swims
  downstream. Which swimmer wins the race?
• A) Ann
• B) Beth
• C) Carly

Time to get across width of river/vertical
component of velocity
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Follow-up Question

• Heather wants to swim across a flowing river in
  such a way that she ends up on the opposite side
  directly opposite her starting point. She should
  therefore aim.
• 1) upstream
• 2) downstream
• 3) directly across

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Question
A seagull flies through the air with a velocity
of 9 m/s if there were no wind. Assume it can
only make the same effort while flying in wind.
It makes a daily round-trip to a rock-island one
km in the sea from the shore. Compare the time it
takes for the seagull to fly on a windy day to
that of a calm day? Assume that the wind is
blowing towards the shore at 5 m/s constantly on
the windy day. 1. The round-trip time is the
same with/without the wind 2. The round trip
time is always larger with the wind 3. It is not
possible to calculate this

• Seagulls relative velocity to wind or earth is
  the same on the windless day
• Calculate seagulls relative velocity to earth
  for the windy day
• It is 14 m/s in one direction and 4 m/s in the
  other
• Calculate times of journey for the roundtrips and
  check the answer

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Lecture 5, Pre-Flight 1314

• For an object traveling at constant speed
• 1 the acceleration must be zero
• 2 the acceleration could be either zero or
  non-zero
• 3 the acceleration can never be zero

If the motion is in a straight line acceleration
is zero If the motion is curved, the speed may
be constant but the direction is changing
continuously Velocity is a vector - in this case
acceleration is non zero
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Lecture 4, Pre-Flight 78
You and a friend are standing on level ground,
each holding identical baseballs. At exactly the
same time, and from the same height, you drop
your baseball without throwing it while your
friend throws her baseball horizontally as hard
as she can. Which ball hits the ground first? 1.
Your ball 2. Your friends ball 3. They both hit
the ground at the same time
The starting speed and the acceleration in the
vertical dimension of both the balls is the same.
The motion in horizontal dimension is independent
of the vertical motion.
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Lecture 4, Pre-Flight 910
A flatbed railroad car is moving along a track at
constant velocity. A passenger at the center of
the car throws a ball straight up. Neglecting
air resistance, where will the ball land? 1.
Forward of the center of the car 2. At the center
of the car 3. Backward of the center of the car
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Lecture 4, Pre-Flight 1112
You drop a ball in an elevator which is
descending at a constant velocity. The
acceleration of the ball is 1. Greater than g 2.
Equal to g 3. Less than g