Motion in two and three dimensions

Motion in two and three dimensions
Luis Anchordoqui!
Position vector
Position vector of a particle is a vector drawn from origin
of a coordinate system to location of a particle !
For a particle in y-x plane at point with coordinates (x,y) !
~r = x ˆı + y |ˆ
Displacement vector
Particle's change in position is displacement vector !
~r = ~r2
~r = (x2
x1 ) ˆı + (y2
~r1
y1 ) |ˆ =
x ˆı +
y ˆı
Velocity vector
~r
Average velocity vector ! ~vav =
t
Instantaneous velocity vector ! ~v = lim
t! 0
~r
d~r
=
t
dt
dx
dy
~v =
ˆı +
|ˆ = vz ˆı , + vy |ˆ
dt
dt
v=
q
vx2
+
vy2

vy
✓ = arctan
vx
Luis Anchordoqui
Velocity of a sailboat
A sailboat has coordinates (x1 , y1 ) = (130 m, 205 m) at t! 1 = 60 s
Two minutes later at time t2 has coordinates!(x2 , y2 ) = (110 m, 218 m)
(a) Find average velocity for this two minute interval
~vav = 0.167 m/s ˆı + 0.108 m/s |ˆ
|~vav | = 0.199 m/s
✓ = 147
Relative Velocity
We use coordinate axes that are attached to reference frames
to make position measurements
A coordinate axis is said to be attached to a reference frame
if coordinate axis is at rest relative to reference frame
Example: Midair refueling
Each plane is nearly at rest relative to each other
though both are moving with relative large velocities relative to Earth !
Relative Velocity (Cont’d)
If particle p moves with velocity ~
vpA relative to a reference frame A
that is in turn moving with velocity ~
vAB relative to a reference frame B
velocity ~
vpB of particle relative to reference frame
by ☛
B is related to ~vpA &!
~vAB
~vpB = ~vPA + ~vAB
Example
If a person p is on a railroad car C
that is moving with velocity
~vCG
relative to ground G
and person is walking with velocity pC relative to car
~v
then velocity of person relative to G is vector sum of these two velocities
~vpG = ~vpC + ~vCG
A Flying Plane
A pilot wishes to fly a plane due north relative to ground !
Airspeed of plane is 200 km/h and wind is blowing from west !
to east at 90 km/h. !
(a) In which direction should plane head? !
(b) What is ground speed of plane? !
27
west of north
plane velocity relative to ground is
180 km/h
Acceleration vector
Average acceleration vector!
~aav =
~v
t
Instantaneous acceleration vector!
~v
d~v
~a = lim
=
t!0
t
dt
dx
dy
dz
~v = vxˆı + vy |ˆ + vz kˆ =
ˆı +
|ˆ +
kˆ
dt
dt
dt
~a = axˆı + ay |ˆ + az kˆ
where!
dvx
dvy
dvz ˆ
d2 x
d2 y
d2 z ˆ
~a =
ˆı +
|ˆ +
k =
ˆı + 2 |ˆ + 2 k
2
dt
dt
dt
dt
dt
dt
Go Sox!!!
The 2004 World Series represented 100th time two modern Major League Baseball
teams met to decide championship, and began on October 23
After winning four consecutive games, on October 27 at 10:40 p.m. CT, American
League champion Boston Red Sox defeated National League champion St. Louis
Cardinals to claim 2004 World Series Trophy. It had been 86 years since Boston
last claimed prize by defeating Chicago Cubs in 1918 World Series
With their Series sweep, Boston had finally broken "Curse of the Bambino”
To make championship sweeter, it came on the heels of largest comeback in
postseason MLB history (a 0-3 deficit against archrival New York Yankees in AL
Championship Series).
Position of a thrown baseball by Pedro Martinez is given by
~r = [1.5 m + (12 m/s) t] ˆı + [(16 m/s) t
(4.9 m/s2 ) t2 ] |ˆ
Find its velocity and acceleration as a function of time
~v = (12 m/s) ˆı + [16 m/s
(9.8 m/s2 ) t]ˆ
|
~a = ( 9.8 m/s2 ) |ˆ
Projectile Motion
This type of motion occurs when on object is launched into air and
is allowed to move freely. Initial velocity then has components !
v0x = v0 cos ✓0
v0y = v0 sin ✓0
In absence of air resistance acceleration is constant !
Component x of velocity is constant!
because no horizontal acceleration exists !
ax = 0
y
component of velocity varies with time according to !
ay =
g
Path of a Projectile
Displacements x and y are given by!
y(t) = y0 + v0y t
x(t) = x0 + v0x t
Velocity components!
vx = v0x
vy = v0y
Eliminating variable! t
y(x) = v0y
✓
x
v0x
y(x) = (tan ✓0 ) x
◆
✓
1 2
gt
2
gt
g
2
✓
x
v0x
g
2v02 cos2 ✓0
◆2
◆
x2
Luis Anchordoqui
Horizontal Range of a Projectile
Horizontal range of a projectile can be written in terms
of its initial speed and initial angle above horizontal !
Flight time is obtained by setting y = 0 !
0 = v0y t
1 2
gt
2
t>0
Flight time of projectile is thus !
2v0y
2v0 sin ✓0
T =
=
g
g
To find horizontal range we substitute flight time
in x-equation of motion!
2
2v0
R = v0x T =
sin 0 cos 0
g
Horizontal Range of a Projectile
(Cont’d)
This can be further simplified by using trigonometric identity !
v02
sin 2# = 2 sin # cos # ) R =
sin(2✓0 )
g
Argentine 17 France 12
Argentina rocked tournament hosts France with a sensational 17-12 victory in the
opening match of the 2007 Rugby World Cup in Paris. The inspired Pumas outplayed an
error-ridden and nervous-looking France and took control with a first-half try from fullback Ignacio Corleto. Centre Felipe Contepomi chipped in with 12points to leave the 80,000strong Stade de France crowd stunned. France could only muster four penalties from
struggling fly-half David Skrela. At the kick-off the football went up at a 37 degree
angle with a velocity of 20 m/s. !
Calculate: !
maximum height is
(a) maximum height !
(b) time of travel before football hits the ground !
(c) how far away it hits ground! total distance
(d) the velocity vector at maximum height!
7.35 m
2.45 s
traveled is 39.2
m
At highest point there is no vertical component of velocity & vx = 16m/s
!
(e) acceleration vector at maximum height!
Assume the ball leaves foot at ground level, and ignore air resistance and rotation of ball !
Acceleration vector is same at highest point as it is through out flight:
9.8 m/s downward
A cup in the air
A delighted physics graduate throws her cap into the air with an
initial velocity of 24.5 m/s at 36.9 degrees above the horizontal. !
The cap is later caught by another student. !
Find !
(a) the total time the cap is in the air! t = 3.00 s
x = 58.8 m
(b) the horizontal distance traveled !
Position of a thrown
To catch a thief
A police officer chases a master jewel thief across city rooftops They are
both running when they come to a gap between buildings that is 4.00 m
wide and has a drop of 3.00 m!
Thief having studied a little of physics, leaps at 5.00 m/s at an angle of
45 degrees above horizontal, and clears gap easily !
Police officer did not study physics and thinks he should maximize his
horizontal velocity, so he leaps horizontally at 5.00 m/s. !
(a)  Does police clear gap !
(b) By how much does thief clear gap !
Police officer fails to make it across gap between buildings
By 0.31 m
Catapults date from thousands of years ago, and were used historically to
launch everything from stones to horses. During a battle on what is now
Bavaria, inventive artillerymen from the united German clans launched giant!
spaetzle from their catapults towards a Roman fortification whose walls were
8.5 m high. !
The catapults launched the spaetzle projectiles from a height of4.00 m
above the ground and a distance of 38.0 m from the walls of the
fortification at an angle of 60 degrees above the horizontal. If the
projectiles were to hit the top of the wall, splattering the Roman soldiers
atop the wall with pulverized pasta !
(a) what launch speed was necessary? !
(b) How long where the spaetzle in the air? !
(c) At what speed did the projectiles hit the wall? !
Ignore any effects due to air resistance. !
Ranger & Monkey
A park ranger with a tranquilizer dart gun intends to shoot a monkey
hanging from a branch. The ranger points the barrel directly at the monkey,
not realizing that the dart will follow a parabolic path that will pass below the
present position of the creature. The monkey, seeing the gun discharge,
immediately lets go of the branch and drops out of the tree, expecting to avoid
the dart. !
Show that the monkey will be hit regardless of the initial speed of the dart as
long as this speed is great enough for the dart to travel the horizontal distance
to the tree. !
Assume the reaction time of the monkey is negligible. !
Luis Anchordoqui
Ranger & Monkey (Cont’d)
!
What is the minimum initial speed of the dart if it is to hit the
monkey before the monkey hits the ground which is 11.2 m
below the initial position of the monkey, if x = 50 m and
h = 10 m? (Ignore any effects due to air resistance) !
v > 34 m/s
007, flying a constant 215 km/h horizontally in a low-flying
helicopter,wants to drop secret documents into his contact's open car
which is traveling155 km/h on a level highway 78.0 m below !
At what angle (to horizontal) should car be in his sights when
packet is released? !
an angle of 50
Luis Anchordoqui
Romeo is chucking pebbles gently up to Juliet's window, and he wants
pebbles to hit window with only horizontal component of velocity !
He is standing at edge of a rose garden 4.5 m below her window and 5
m from base of wall!
How fast are the pebbles going when
they hit the window? !
v = 5.2 m/s
A rescue plane wants to drop supplies to isolated mountain climbers on a rocky
ridge 235 m below. If plane is traveling horizontally with a speed of 250 km/h,
how far in advance of recipients (horizontal distance) must goods be dropped? !
x = 481 m
Suppose, instead, that plane releases supplies a horizontal distance of425 m in
advance of mountain climbers. !
What vertical velocity (up or down) should supplies be given so that they arrive
precisely at climbers position? !
With what speed do supplies land in latter case? !
vy0 =
v=
q
8.37 m/s
vx2 + vy2 = 97.4 m/s
At
t = 0 a batter hits a baseball with an initial speed of 32 m/s
angle to horizontal. An outfielder is 85 m from batter at
t = 0,
at a 55
and as
seen from home plate, line of sight to outfielder makes a horizontal
angle of
22
which plane in which ball moves
What speed and direction must fielder take in order to catch ball at same
height from which it was struck?
Give angle with respect to outfielder's line of sight to home plate
Average velocity is 7 m/s at an angle of
outfielder’s line of sight to home plate
97 relative to

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