#### Transcript Document

```PHSX213 class
• I hope you had a good weekend.
– I spent a lot of Friday evening annoyed that my
car seems to obey Newton’s 1st Law when I was
turning on an icy corner !
• HW1 is due now.
• Questions from last time ? (1-d kinematics)
• Many of you are now signed up for the
online access – please do this as soon as
you can.
– HW2 is available (only available with
– Submissions
• Kinematics in 2-d (Chapter 4)
– Projectile motion
– Uniform Circular Motion
Mon. Jan. 31st
1
Position vector for a particle
Mon. Jan. 31st
2
Mon. Jan. 31st
3
A ball is thrown upward at a 45° angle. In the absence of
air resistance, the ball follows a
A. tangential curve.
B. parabolic curve.
C. sine curve.
D. linear curve.
Mon. Jan. 31st
4
A ball is thrown upward at a 45° angle. In the absence of
air resistance, the ball follows a
A. tangential curve.
B. parabolic curve.
C. sine curve.
D. linear curve.
Mon. Jan. 31st
5
Check-Point 0
You are throwing a ball straight up in the air.
At the highest point, the ball’s
A. velocity and acceleration are zero.
B. velocity is nonzero but its acceleration is zero.
C. acceleration is nonzero, but its velocity is zero.
D. velocity and acceleration are both nonzero.
Mon. Jan. 31st
6
Check-Point 0
You are throwing a ball straight up in the air.
At the highest point, the ball’s
A. velocity and acceleration are zero.
B. velocity is nonzero but its acceleration is zero.
C. acceleration is nonzero, but its velocity is zero.
D. velocity and acceleration are both nonzero.
Mon. Jan. 31st
The ball reaches its highest point when its
velocity is zero. The acceleration due to
gravity is constant and never zero.
7
Check-Point 1
A person standing at the edge of a cliff
throws one ball straight up and another ball
straight down at the same initial speed. Neglecting
air resistance, the ball to hit the
ground below the cliff with the greater speed
is the one initially thrown :
A. upward.
B. downward.
C. neither—they both hit at the same speed.
Mon. Jan. 31st
8
Check-Point 1
A person standing at the edge of a cliff
throws one ball straight up and another ball
straight down at the same initial speed. Neglecting
air resistance, the ball to hit the
ground below the cliff with the greater speed
is the one initially thrown :
On its descent, an object
with initial velocity, v, has a
velocity of –v, when it
A. upward.
reaches the height from
B. downward.
which it was thrown
C. neither—they both hit at the same speed.
Mon. Jan. 31st
9
Kinematics: General Case (3-D)
• Reference frame
– Origin, (x, y, z)-axes
•
•
•
•
r→=
^
^
^
Position Vector,
xi+yj+zk
Displacement, ≡ D →
r =→
r2 – →
r1
→
→
Instantaneous Velocity, v ≡ d r /dt
→
→
→
2
Instantaneous Acceleration, a ≡ d v/dt ≡ d r /dt2
Mon. Jan. 31st
10
Let’s look in more detail at what
this 3-D stuff means
Mon. Jan. 31st
11
Projectile Motion
• (Neglect air resistance)
• Launch a projectile at angle, q0, to
the horizontal, and with initial speed,
v0
• Horizontal component of the
velocity, vx , experiences NO
acceleration (ie. ax = 0).
• Vertical component of the velocity,
vy , has acceleration due to gravity
(ie. ay = -g ).
• Projectile motion can be analyzed by
considering the horizontal and
vertical as independent of each other.
• Range = horizontal distance R.
Mon. Jan. 31st
12
Check-Point 2
Consider the situation depicted here. A gun
is accurately aimed at a dangerous criminal
hanging from the gutter of a building. The
target is well within the gun’s range, but the
instant the gun is fired and the bullet moves
with a speed vo, the criminal lets go and
drops to the ground. What happens? The
bullet :
• 1. hits the criminal regardless of the
value of vo.
• 2. hits the criminal only if vo is large
enough.
• 3. misses the criminal.
Mon. Jan. 31st
13
Check-Point 2
Consider the situation depicted here. A gun
is accurately aimed at a dangerous criminal
hanging from the gutter of a building. The
target is well within the gun’s range, but the
instant the gun is fired and the bullet moves
with a speed vo, the criminal lets go and
drops to the ground. What happens? The
bullet :
• 1. hits the criminal regardless of the
value of vo.
• 2. hits the criminal only if vo is large
enough.
• 3. misses the criminal.
Why don’t you convince
yourself for next time.
Mon. Jan. 31st
14
A hunter points his rifle directly at a coconut that he wishes
to shoot off a tree. It so happens that the coconut falls from
the tree at the exact instant the hunter pulls the trigger.
Consequently,
A. the bullet passes above the coconut.
B. the bullet passes beneath the coconut.
C. the bullet hits the coconut.
D. A situation similar to this wasn’t
discussed in Chapter 4.
Mon. Jan. 31st
15
A hunter points his rifle directly at a coconut that he wishes
to shoot off a tree. It so happens that the coconut falls from
the tree at the exact instant the hunter pulls the trigger.
Consequently,
A. the bullet passes above the coconut.
B. the bullet passes beneath the coconut.
C. the bullet hits the coconut.
D. A situation similar to this wasn’t
discussed in Chapter 4.
Mon. Jan. 31st
16
Projectile Demo
• Car on track with steel ball.
Mon. Jan. 31st
17
Projectile Problems
• Projectile (eg. punted football) launched with
speed, v0, at launch angle, q.
• Time to reach its highest point ?
• Coordinates of highest point ?
• Time to hit ground ?
• Range ?
• Velocity components just before landing ?
• What launch conditions for maximum range ?
Mon. Jan. 31st
18
Check-Point 3
A battleship simultaneously fires two shells at
enemy ships. If the shells follow the parabolic
trajectories shown, which ship gets hit first?
A. A
B. B
C. both at the same time
Mon. Jan. 31st
19
Check-Point 3
A battleship simultaneously fires two shells at
enemy ships. If the shells follow the parabolic
trajectories shown, which ship gets hit first?
A. A
B. B
C. both at the same time
Mon. Jan. 31st
The time a projectile
spends in the air is twice
the time taken to fall from
its maximum height.
20
Uniform Circular Motion
• An object traveling with
constant speed in a
circular path IS
accelerating (because the
velocity is not constant)
• The centripetal
acceleration is given by :
• a = v2/r
• Period, T, and revolution
frequency definitions.
Mon. Jan. 31st
21
Centripetal acceleration
derivation
• Note the MINUS sign.
Mon. Jan. 31st
22
On Wednesday
• Relative Motion (end of chapter 4)
• Start Newton’s Laws (chapter 5)
Mon. Jan. 31st
23
```