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Physics 151: Lecture 5, Pg 1
Announcements:Announcements: Physics Learning Resource Center Open,
– > room P207-COpen 9am - 5 pm Monday - FridayHours also listed on www.phys.uconn.edu
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 2
Projectile motion
Uniform circular motion
text sections 4.1-4.5
Physics 151: Lecture 5Physics 151: Lecture 5Today’s AgendaToday’s Agenda
(Chapter #4)(Chapter #4)
Physics 151: Lecture 5, Pg 3
ACT -1ACT -1Question from past Exam-1Question from past Exam-1
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.
Two balls, projected at different times so they don’t collide, have trajectories A and B, as shown.
Physics 151: Lecture 5, Pg 4
Review : Kinematics in 3D (2D)Review : Kinematics in 3D (2D) Motion of objects in 3-D under constant acceleration
(problem in projectile motion) :
y
x
-> independence of x- and y- components :
yy (t)(t)
xx (t)(t)
aayy= -g= -g
aaxx=0=0
yyoo
xxoo
ttoo= 0= 0
ttvx
vy
vvoovoy
vox
y = y0 + v0y t - 1/ 2 g t2
vy = v0y - g t
x = vx tvx = v0x
x-components (a=0)
y-components (a=-g)
Animation (v) Animation (a)
Physics 151: Lecture 5, Pg 5
Review : Kinematics in 3D (2D) (cont.)Review : Kinematics in 3D (2D) (cont.)
y
x
yy (t)(t)
xx (t)(t)
aayy= -g= -g
aaxx=0=0
yyoo
xxoo
ttoo= 0= 0
ttvx
vy
vvoovoy
vox
How many parameter determine projectile motion ?
at to = 0: xo, yo , vox, voy
at t = t : x(t) , y(t), vx, vy
9 variablesplus a=g {
we have 4 equations !
}4 independent variables !
Physics 151: Lecture 5, Pg 6
Projectile MotionProjectile Motion
Typical Questions,1. x: How far will it go ?
2. y: How high will it be at some distance ?
3. t: How long until it hits ?
4. : At what angle should I start ?
5. v0: How fast must I start ?
v0
g
Physics 151: Lecture 5, Pg 7
Projectile MotionProjectile Motion
Useful Things to Know
If projectile begins and ends at same height, maximum distance is achieved for = 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.
v0
g
Physics 151: Lecture 5, Pg 8
Projectile Motion / Example ProblemProjectile Motion / Example Problem
v0
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: = 15° or 75° which gets there first ?
Physics 151: Lecture 5, Pg 9
Problem 3Problem 3 (correlated motion of 2 objects in 3-D) (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.
t = t1
y
x
vv00
t = 0
t = 0TARGET
PROJECTILE
( A ) MISS ( B ) HIT ( C ) CAN’T TELL
Would the projectile now miss or hit the target ?
Physics 151: Lecture 5, Pg 10
ReviewReview( displacement, velocity, acceleration )( displacement, velocity, acceleration )
Velocity :
Acceleration :
a = dvv / dt
3-D Kinematics : vector equations:
rr = rr(t) vv = drr / dt aa = d2rr / dt2
v = drr / dt
vav = rr / t
aav = vv / t
vv2
vv-v-v1
vv2
y
x
path
vv1
Physics 151: Lecture 5, Pg 11
General 3-D motion with non-zero acceleration:General 3-D motion with non-zero acceleration:
Uniform Circular Motion is one specific case of this :
aa
vpath
t
aa
aa
aa = 0
because either or both:
-> change in magnitude of v
-> change in direction of v
aa = 0
aa = 0
aaaaaa= +
Animation
Physics 151: Lecture 5, Pg 12
Uniform Circular MotionUniform Circular Motion
What does it mean ?
How do we describe it ?
What can we learn about it ?
See text: 4-4
Physics 151: Lecture 5, Pg 13
What is Uniform Circular Motion (UCM) ?What is Uniform Circular Motion (UCM) ?
Motion in a circle with:
Constant Radius R
Constant Speed v = |vv|
acceleration ?
R
vv
x
y
(x,y)
See text: 4-4
= 0aa
aa
= const.aa
Physics 151: Lecture 5, Pg 14
How can we describe UCM?How can we describe UCM?
In general, one coordinate system is as good as any other: Cartesian:
» (x,y) [position]» (vx ,vy) [velocity]
Polar:» (R,) [position]» (vR ,) [velocity]
In UCM: R is constant (hence vR = 0). (angular velocity) is constant. Polar coordinates are a natural way to describe UCM!Polar coordinates are a natural way to describe UCM!
RR
vv
x
y
(x,y)
See text: 4-4
Physics 151: Lecture 5, Pg 15
Polar Coordinates: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.
For one complete revolution:
2R = Rc
c = 2
has period 2.RR
vv
x
y
(x,y)s
See text: 4-4
1 revolution = 21 revolution = 2radiansradians
Physics 151: Lecture 5, Pg 16
Polar Coordinates...Polar Coordinates...
In Cartesian co-ordinates we say velocity dx/dt = v. x = vt
In polar coordinates, angular velocity d/dt = . = t has units of radians/second.
Displacement s = vt.
but s = R = Rt, so:RR
vv
x
y
st
v = R
Physics 151: Lecture 5, Pg 17
Period and FrequencyPeriod and Frequency Recall that 1 revolution = 2 radians
frequency (f) = revolutions / second (a)angular velocity () = radians / second (b)
By combining (a) and (b) = 2 f
Realize that:period (T) = seconds / revolutionSo T = 1 / f = 2/
RR
vv
s
= 2 / T = 2f
vv
R
s
Physics 151: Lecture 5, Pg 18
Lecture 5, Lecture 5, ACT 2ACT 2Uniform Circular MotionUniform 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 19
Acceleration in UCM:Acceleration in UCM:
This is called This is called Centripetal Acceleration. Now let’s calculate the magnitude:
vv1vv2
vv
vv2
vv1RR
RR
vv
RR
Similar triangles:
But R = vt for small t
So: vv
v tR
vt
vR
2
avR
2
Physics 151: Lecture 5, Pg 20
Lecture 5, Lecture 5, ACT 2ACT 2Uniform Circular MotionUniform 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 21
Recap for today:Recap for today: Recap of Projectile Motion
Introduce Uniform Circular Motion
Reading assignment for MondayReread Circular Motion: Ch 4.4-4, pp.91-95Read Relative Velocity: Ch 4.5, pp.95-99
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