Transcript Physics 131: Lecture 6 Notes

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NOTE:
Homework #1 (due this Fri. 9/8 by 5:00 pm EST
on WebAssign)
Homework #2 (due next Fri. 9/15 by 5.00 pm)
Physics 151: Lecture 5, Pg 1
Physics 151: Lecture 5
Today’s Agenda
(Chapter #4)
Projectile motion
Uniform circular motion
text sections 4.1-4.5
Physics 151: Lecture 5, Pg 2
ACT -1
Question from past Exam-1
Two balls, projected at different
times so they don’t collide, have
trajectories A and B, as shown.
Which statement is true.
1. Initial speed of ball B must be greater than that of ball A.
2. Ball A is in the air for a longer time than ball B.
3. Ball B is in the air for a longer time than ball A.
4. Ball B has a greater acceleration than ball A.
5. None of above.
Physics 151: Lecture 5, Pg 3
Review : Kinematics in 3D (2D)

Motion of objects in 3-D under constant acceleration
(problem in projectile motion) :
-> independence of x- and y- components :
ax=0
ay= -g
y
t
y (t)
v
voy o

yo
vox
t o= 0
xo
vx
vy
x (t)
y-components (a=-g)
y = y0 + v0y t - 1/ 2 g t2
vy = v0y - g t
x
x-components (a=0)
Animation (v)
x = vx t
vx = v0x
Animation (a)
Physics 151: Lecture 5, Pg 4
Review : Kinematics in 3D (2D) (cont.)
ax=0
ay= -g

y
t
y (t)
v
voy o

yo
vox
t o= 0
xo
vx
vy
x (t)
x
How many parameter determine projectile motion ?
9 variables
plus a=g
at to = 0:
{ at t = t :
xo ,
yo , vox, voy
x(t) , y(t), vx, vy
we have 4 equations !
} 4 independent variables !
Physics 151: Lecture 5, Pg 5
Projectile Motion
g
v0
Q
Typical Questions,
1. Dx: How far will it go ?
2. Dy: How high will it be at some
distance ?
3. t: How long until it hits ?
4. Q: At what angle should I start ?
5. v0: How fast must I start ?
Physics 151: Lecture 5, Pg 6
Projectile Motion
g
v0
Q
Useful Things to Know



If projectile begins and ends at same height, maximum distance is
achieved for Q = 45°. (prove it)
x distance is same for angles about 45° if everything else remains the
same.
Time in flight depends on y equation if no barriers other than the earth
interrupt the flight path.
Physics 151: Lecture 5, Pg 7
Projectile Motion / Example Problem
v0
Q
g
UConn football team wants to complete a 45m pass
(about 50 yards). Our qb can throw the ball at 30 m/s.
At what angle must he throw the ball to get it there ?
SOLUTION:
Q = 15°
or
75°
which gets there first ?
Physics 151: Lecture 5, Pg 8
Problem 3
(correlated motion of 2 objects in 3-D)

Suppose a projectile is aimed at a target at rest somewhere
above the ground as shown in Fig. below. At the same time that
the projectile leaves the cannon the target falls toward ground.
Would the projectile now miss or hit the target ?
( A ) MISS
( B ) HIT
( C ) CAN’T TELL
t=0
TARGET
y
v0
t=0
t = t1

PROJECTILE
x
Physics 151: Lecture 5, Pg 9
Review
( displacement, velocity, acceleration )

3-D Kinematics : vector equations:
r = r(t)
v = dr / dt
v2
y
v1
a = d2r / dt2
Velocity :
path
vav = Dr / Dt
v2
Dv
-v1
v = dr / dt
x
Acceleration :
aav = Dv / Dt
a = dv / dt
Physics 151: Lecture 5, Pg 10
General 3-D motion with non-zero acceleration:
v
path
a
t
a = a + a
a
a
a =0
because either or both:
v
-> change in magnitude of
-> change in direction of
v
a
=0
a
=0
Animation

Uniform Circular Motion is one specific case of this :
Physics 151: Lecture 5, Pg 11
See text: 4-4
Uniform Circular Motion

What does it mean ?

How do we describe it ?

What can we learn about it ?
Physics 151: Lecture 5, Pg 12
See text: 4-4
What is Uniform Circular Motion (UCM) ?

Motion in a circle with:
y
Constant Radius R
v
(x,y)
a
 Constant Speed v = |v|
R
x
acceleration ?
a
=0
a
= const.
Physics 151: Lecture 5, Pg 13
See text: 4-4
How can we describe UCM?

In general, one coordinate system is as good as any other:
 Cartesian:
» (x,y) [position]
y
» (vx ,vy) [velocity]
v
 Polar:
(x,y)
» (R,) [position]
R

» (vR ,) [velocity]
x

In UCM:
 R is constant (hence vR = 0).
  (angular velocity) is constant.
 Polar coordinates are a natural way to describe UCM!
Physics 151: Lecture 5, Pg 14
See text: 4-4
Polar Coordinates:

The arc length s (distance along the circumference) is related to
the angle in a simple way:
s = R, where  is the angular displacement.
 units of  are called radians.
y

For one complete revolution:
2R = Rc
 c = 2
 has period 2.
v

(x,y)
R s
x
1 revolution = 2radians
Physics 151: Lecture 5, Pg 15
Polar Coordinates...



In Cartesian co-ordinates we say velocity dx/dt = v.
 x = vt
In polar coordinates, angular velocity d/dt = .
  = t
y
 has units of radians/second.
v
Displacement s = vt.
R
but s = R = Rt, so:
t
v = R
s
x
Physics 151: Lecture 5, Pg 16
Period and Frequency



Recall that 1 revolution = 2 radians
frequency (f) = revolutions / second
angular velocity () = radians / second
(a)
(b)
By combining (a) and (b)
 = 2 f
Realize that:
period (T) = seconds / revolution
So T = 1 / f = 2/
v
v
R
R
s
s

 = 2 / T = 2f
Physics 151: Lecture 5, Pg 17
Lecture 5, ACT 2
Uniform Circular Motion

A fighter pilot flying in a circular turn will pass out if the
centripetal acceleration he experiences is more than about
9 times the acceleration of gravity g. If his F18 is moving
with a speed of 300 m/s, what is the approximate diameter
of the tightest turn this pilot can make and survive to tell
about it ?
Physics 151: Lecture 5, Pg 18
Acceleration in UCM:


This is called Centripetal Acceleration.
Now let’s calculate the magnitude:
Dv
v2
v2
R
DR
v1
Similar triangles:
v1
Dv DR

v
R
But DR = vDt for small Dt
Dv vDt

So:
v
R
Dv v 2

Dt
R
v2
a
R
Physics 151: Lecture 5, Pg 19
Lecture 5, ACT 2
Uniform Circular Motion

A fighter pilot flying in a circular turn will pass out if the centripetal
acceleration he experiences is more than about 9 times the
acceleration of gravity g. If his F18 is moving with a speed of 300
m/s, what is the approximate diameter of the tightest turn this pilot
can make and survive to tell about it ?
(a) 20 m
(b) 200 m
(c) 2000 m
(d) 20,000 m
Physics 151: Lecture 5, Pg 20
Recap for today:

Recap of Projectile Motion

Introduce Uniform Circular Motion

Reading assignment for Monday
Reread Circular Motion: Ch 4.4-4, pp.91-95
Read Relative Velocity: Ch 4.5, pp.95-99

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Homework #1 (due this Fri. 9/8 by 5:00 pm EST on
WebAssign) Problems from Chapter 1 and 2
Homework #2 (due next Fri. 9/15 by 5.00 pm)
Problems from Chapter 3 and 4
Physics 151: Lecture 5, Pg 21