CHAPTER 3

Two-Dimensional Motion and Vectors

Representations:

x

y

(x, y)

(x, y) (r, !)

VECTOR quantities:

Vectors have magnitude and direction.

Other vectors: velocity, acceleration, momentum, force …

Vector Addition/Subtraction

• 2nd vector begins at end of first vector

• Order doesn’t matter

Vector addition

Vector subtraction

A – B can be interpreted as A+(-B)

Vector Components

Cartesian components are projections along the x- and y-axes

Ax= Acos!

Ay = Asin!

Going backwards,

A = Ax2+ Ay

2and ! = tan

"1Ay

Ax

Example 3.1a

The magnitude of (A-B) is :

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

Example 3.1b

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

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

Example 3.1c

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

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

Example 3.2

Alice and Bob carry a bottle of wine to a picnic site. Alice carries the bottle 5 miles due east, and Bob carries the bottle another 10 miles 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

Arcsin, Arccos and Arctan: Watch out!

same sine

same

cosinesame

tangent

Arcsin, Arccos and Arctan functions can yield wrong angles if x or y are negative.

2-dim Motion: Velocity

Graphically,

v = "r / "t

It is a vector

(rate of change of position)

Trajectory

Multiplying/Dividing Vectors by

Scalars, e.g. "r/"t

• 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

Principles of 2-d Motion

• 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)

Projectile Motion

• X-motion is at constant velocity ax=0, vx=constant

• Y-motion is at constant accelerationay=-g

Note: we have ignored

• air resistance

• rotation of earth (Coriolis force)

Projectile Motion

Acceleration

is constant

Pop and Drop Demo

The Ballistic Cart

Finding Trajectory, y(x)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,xt

y = v0,yt !1

2gt2

t = x / v0,x

y =v0,y

v0,xx !

1

2

g

v0,x2x2

Example 3.3

An airplane drops food to two starving hunters. The plane is flying at an altitude of 100 m and with a velocity of 40.0 m/s.

How far ahead of the hunters should the plane release the food?

X181 m

h

v0

Example 3.4a

h

D!

v0

The Y-component of v at A is :

a) <0b) 0c) >0

Example 3.4b

h

D!

v0

a) <0b) 0c) >0

The Y-component of v at B is

Example 3.4c

h

D!

v0

a) <0b) 0c) >0

The Y-component of v at C is:

Example 3.4d

h

D!

v0

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

The speed is greatest at:

Example 3.4e

h

D!

v0

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

The X-component of v is greatest at:

Example 3.4f

h

D!

v0

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

The magnitude of the acceleration is greatest at:

Range Formula

• Good for when yf = yi

x = vi,xt

y = vi,yt !1

2gt2= 0

t =2vi,y

g

x =2vi,xvi,y

g=2vi

2cos" sin"

g

x =vi2

gsin2"

Range Formula

• Maximum for !=45°R =vi2

gsin2!

Example 3.5a

100 m

A softball leaves a bat with an initial velocity of 31.33 m/s. What is the maximum distance one could expect the ball to travel?

Example 3.6

299 m

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

h

D!

v0

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

B. Considering the effects of gravity, at what angle should the hunter aim the arrow relative to the ground?

Example 3.7, Shoot the Monkey

!=Arctan(h/L)=26.6°

A hunter is a distance L = 40 m from a tree in which a monkey is perched a height h=20 m above the hunter. The hunter shoots an arrow at the monkey. However, this is a smart monkey who lets go of the branch the instant he sees the hunter release the arrow. The initial velocity of the arrow is v = 50 m/s.

Solution:Must find v0,y/vx 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

2gt2

ymonkey = h !1

2gt2

h !1

2gt2= v0,yt !

1

2gt2

h = v0,yt = v0,yL

vx,

v0,y

vx=h

L

Shoot the Monkey DemoRelative velocity

• 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)

Example 3.8

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

A plane that is capable of traveling 200 m.p.h. flies 100 miles into a 50 m.p.h. wind, then flies back with a 50 m.p.h. tail wind.

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

Relative velocity in 2-d

• Sum velocities as vectors

• velocity relative to ground= velocity relative to medium + velocity of medium.

vbe = vbr + vre

boat wrt river

river wrt earth

Boat wrt earth

2 Cases

pointed perpendicularto stream

travels perpendicularto stream

Example 3.9

An airplane is capable of moving 200 mph in still air. The plane points directly east, but a 50 mph wind from the north distorts his course.

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

206.2 mph14.0 deg. south of east

Example 3.10

An airplane is capable of moving 200 mph in still air. A wind blows directly from the North at 50 mph. The airplane accounts for the wind (by pointing the plane somewhat into the wind) and flies directly east relative to the ground.

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

193.6 mph14.5 deg. north of east

Example 3.11a

Three airplanes, A, B and C, with identical air speeds fly from Williamston, MI, towards Tallahassee, FL, which is directly south. “A” flies on Monday when there is a strong wind from the west. “A” aims the plane south but is blown off course. “B” also leaves Monday, but aims a bit into the wind and lands in Tallahassee. “C”flies on Tuesday, a calm and windless day, and flies directly to Tallahassee. Which plane(s) has(have) the HIGHEST ground speed?

A) AB) BC) CD) A and BE) B and C

Example 3.11b

Three airplanes, A, B and C, with identical air speeds fly from Williamston, MI, towards Tallahassee, FL, which is directly south. “A” flies on Monday when there is a strong wind from the west. “A” aims the plane south but is blown off course. “B” also leaves Monday, but aims a bit into the wind and lands in Tallahassee. “C”flies on Tuesday, a calm and windless day, and flies directly to Tallahassee. Which plane(s) has(have) the LOWEST ground speed?

A) AB) BC) CD) A and BE) B and C