Chapter3.Key

7/27/2019 Chapter3.Key

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Two-Dimensional Motion andVectors

x

y

(x, y)

(x, y) (r, !)

Vectors have magnitudeand direction.

Other vectors: velocity,acceleration, momentum,force …

• 2nd vector begins atend of first vector

• Order doesn’t matter

Vector addition

Vector subtraction

A – B can be interpretedas A+(-B)

Cartesian components areprojections along the x-and y-axes

A x = Acos!

A y = Asin!

Going backwards,

A = A x2 + A y2 and ! = tan"1A y

A x

The magnitude of (A-B) is :

a) <0b) =0c) >0

The x-component of (A-B) is:

a) <0b) =0c) >0



The y-component of (A-B) > 0

a) <0b) =0c) >0

Alice and Bob carry a bottle of wine to apicnic site. Alice carries the bottle 5 miles dueeast, and Bob carries the bottle another 10miles traveling 30 degrees north of east.Carol, who is bringing the glasses, takes a

short cut and goes directly to the picnic site.

How far did Carol walk?What was Carol’s direction?

14.55 miles, at 20.10 degrees Alice

BobCarol

same sine

same

cosinesame

tangent

Arcsin, Arccos and Arctan functions can yield

wrong angles if x or y are negative.

Graphically,

v = "r / "t

It is a vector

(rate of change of position)

Trajectory

• Vector multiplied/divided by scalar is a vector• Magnitude of new vector is magnitude of

orginal vector multiplied/divided by |scalar|

• Direction of new vector same as original vector

• X- and Y-motion are independent

• Two separate 1-d problems• To get trajectory (y vs. x)

1. Solve for x(t) and y(t)2. Invert one Eq. to get t(x)3. Insert t(x) into y(t) to get y(x)



• X-motion is at constant velocityax=0, v x=constant

• Y-motion is at constant accelerationay=-g

Note: we have ignored

• air resistance

• rotation of earth (Coriolis force)

Acceleration

is constant

1. Write down x(t)

2. Write down y(t)

3. Invert x(t) to find t(x)

4. Insert t(x) into y(t) to get y(x)

Trajectory is parabolic

x = v0, x t

y = v0, yt !1

2gt

2

t = x / v0, x

y =v0, y

v0, x

x !1

2

g

v0, x2

x2

An airplane drops food to

two starving hunters. Theplane is flying at an altitudeof 100 m and with a velocityof 40.0 m/s.

How far ahead of thehunters should the planerelease the food?

X181 m

h

v0



h

D!

v0

The Y-component of v at A is :

a) <0b) 0c) >0

h

D!

v0

a) <0b) 0c) >0

The Y-component of v at B is

h

D!

v0

a) <0

b) 0c) >0

The Y-component of v at C is:

h

D!

v0

a) A

b) Bc) Cd) Equal at all points

The speed is greatest at:

h

D!

v0

a) Ab) Bc) Cd) Equal at all points

The X-component of v is greatest at:

h

D!

v0

a) Ab) Bc) Cd) Equal at all points

The magnitude of the acceleration is greatest at:



• Good for when yf = yi

x = vi, xt

y = vi, yt !1

2gt

2= 0

t =2vi, y

g

x =2vi, xvi, y

g=

2vi2cos" sin"

g

x =vi2

gsin2"

• Maximum for !=45° R =

vi2

gsin2!

100 m

A softball leaves a bat with aninitial velocity of 31.33 m/s. Whatis the maximum distance one couldexpect the ball to travel?

299 m

A cannon hurls a projectile which hits a target located on acliff D=500 m away in the horizontal direction. The cannonis pointed 50 degrees above the horizontal and the muzzlevelocity is 100 m/s. Find the height h of the cliff?

h

D!

v0

A. If the arrow traveled with infinite speed on astraight line trajectory, at what angle should thehunter aim the arrow relative to the ground?

B. Considering the effects of gravity, at what angleshould the hunter aim the arrow relative to theground?

!=Arctan(h/L)=26.6°

A hunter is a distance L = 40 m from a tree in which amonkey is perched a height h=20 m above the hunter. Thehunter shoots an arrow at the monkey. However, this is a

smart monkey who lets go of the branch the instant hesees the hunter release the arrow. The initial velocity ofthe arrow is v = 50 m/s.

Must find v 0,y/v x in terms of h and L

1. Height of arrow

2. Height of monkey

3. Require monkey and arrow to be at same place

Aim directly at Monkey!

yarrow = v0, yt !1

2

gt 2

ymonkey = h !1

2gt

2

h !1

2gt 2 = v0, yt !

1

2gt 2

h = v0, yt = v0, y

L

v x,

v0, y

v x=

h

L



• Velocity always defined relative to reference frame.

All velocities are relative• Relative velocities are calculated by vector addition/

subtraction.

• Acceleration is independent of reference frame

• For high v ~c, rules are more complicated (Einstein)

1.067 hours = 1 hr. and 4 minutes187.4 mph

A plane that is capable of traveling 200 m.p.h. flies100 miles into a 50 m.p.h. wind, then flies back witha 50 m.p.h. tail wind.

How long does the trip take?What is the average speed of the plane for thetrip?

• Sum velocities as vectors

• velocity relative toground= velocity relative to

medium + velocity ofmedium.

v be = v br + v re

boat wrtriver

river wrt

earthBoat wrtearth

pointed perpendicularto stream

travels perpendicularto stream

An airplane is capable of moving 200 mph in

still air. The plane points directly east, but a50 mph wind from the north distorts hiscourse.

What is the resulting ground speed?What direction does the plane fly relative tothe ground?

206.2 mph14.0 deg. south of east



An airplane is capable of moving 200 mph in stillair. A wind blows directly from the North at 50mph. The airplane accounts for the wind (bypointing the plane somewhat into the wind) andflies directly east relative to the ground.

What is the plane’s resulting ground speed?In what direction is the nose of the planepointed?

193.6 mph14.5 deg. north of east

Three airplanes, A, B and C, with identical air speeds flyfrom Williamston, MI, towards Tallahassee, FL, which isdirectly south. “A” flies on Monday when there is a strongwind from the west. “A” aims the plane south but is blownoff course. “B” also leaves Monday, but aims a bit into thewind and lands in Tallahassee. “C”flies on Tuesday, a calmand windless day, and flies directly to Tallahassee. Which

plane(s) has(have) the HIGHEST ground speed?

A) AB) BC) C

D) A and BE) B and C

Three airplanes, A, B and C, with identical air speeds flyfrom Williamston, MI, towards Tallahassee, FL, which isdirectly south. “A” flies on Monday when there is a strongwind from the west. “A” aims the plane south but is blownoff course. “B” also leaves Monday, but aims a bit into thewind and lands in Tallahassee. “C”flies on Tuesday, a calmand windless day, and flies directly to Tallahassee. Whichplane(s) has(have) the LOWEST ground speed?

A) AB) BC) C

D) A and BE) B and C

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Chapter3.Key