Motion in Two Dimensions

Galileo’s study of motion included exploration of motion in two dimensions (animations) Vectors are used to describe motion Vectors have magnitude and direction Scalars are simply numbers (i.e. magnitude only) Vectors displacement distance velocity acceleration force Scalars speed Vectors are denote by bold face or arrows

V

or V

 The magnitude of a vector is denoted by plain text or vertical bars Vectors can be graphically represented by arrows note direction and magnitude

V or V

 Phys 250 Ch3 p1

Vector Addition: Graphical Method of

R

•Shift

B

=

A

+

B

parallel to itself until its tail is at the head of

A

, retaining its original length and direction.

•Draw

R

(the resultant) from the tail of

A

to the head of

B.

B B A

+ =

A

=

R

the order of addition of several vectors does not matter

B C B C A D D A C B D

Phys 250 Ch3 p2

A

the order of addition of several vectors does not matter

B C B C A D D A C B D A

Phys 250 Ch3 p3

Vector Subtraction: the negative of a vector points in the opposite direction, but retains its size (magnitude) •

A

-

B = A +(

-B

)

A

-

B

=

A

+ -

B R

-

B

=

A

Phys 250 Ch3 p4

Resolving a Vector replacing a vector with two or more (mutually perpendicular) vectors => components directions of components determined by coordinates or geometry.

A

y q

A

Examples: horizontal and vertical North-South and East-West

A

x Remember basic trig: SOH CAH TOA Phys 250 Ch3 p5 q A A x A y sin q  opposite hypotenuse 

A y C

cos q  adjacent hypotenuse 

A x A

tan q  opposite adjacent

A x

2 +

A y

2 

A

2 

A A x y

Example: Swimming at an angle of 27º from the horizontal, an angelfish ha a velocity

v

with magnitude 25 cm/s. Find the horizontal and vertical components of

v

.

Phys 250 Ch3 p6

Vector Addition by components

R

=

A

+

B

+

C

Resolve vectors into components(A x , A y etc. ) Add like components A x + B x + C x = R x A y + B y + C y = R y The magnitude and direction of the resultant

R

can be determined from its components. Strategy: 1. Draw a sketch and choose a coordinate system. Use graphical method to estimate result 2. Resolve all vectors into components 3. Add all x-components to get the resultant x-component. Add all y components to get the resultant y-component.

4. Determine the magnitude and direction of the resultant vector, as needed.

Phys 250 Ch3 p7

Example: Vector

A

has length 14 cm at 60º with respect to the +x-axis, and vector

B

length 20 at 20º with respect to the +x-axis. What is the resultant of

A

+

B

has Phys 250 Ch3 p8

Relative Velocity Two frames of reference, one “moving relative to the other” A school bus is traveling at 20m/s relative to the crossing guard. A boy on the bus rolls a ball from the back of the bus to the front with a speed of 5m/s relative to the boy.

How fast does the ball go, elative to the crossing guard?

How does this change if the ball is rolled from the front to the back of the bus?

How does this change if the ball is rolled from the front to the back of the bus?

Relative Velocity

V

AC =

V

AB +

V

BC velocity of A relative to C equals velocity of A relative to B plus velocity of B relative to C Phys 250 Ch3 p9

Example: A person can row a boat 5.00 km/hr in still water tries to cross a river whose current is 3.00 km/hr. The boat is pointed straight across the river, but it is carried downstream by the river as the rower rows across.

What is the velocity of the boat relative to land?

How far down stream does the boat land on the opposite shore if the river is 200 km wide?

A small airplane with an airspeed of 200 km/hr is flown directly north by a novice pilot from Columbia to Charlotte. The wind is blowing from northwest to sought east at 28 km/hr. What is the plane’s resultant speed and direction relative to the ground?

Phys 250 Ch3 p10

Kinematics in Two Dimensions Rates of change: average velocity

v

 

r



t

Change in vector  change in components!

instantaneous velocity and acceleration

v a

  lim 

t

 0 

r



t



v

lim 

t

 0 

t

vectors!

v



v x

2 +

v y

2

a



a x

2 +

a y

2 3-D kinematics:

v



v

0 +

a

t

r



r

0 +

v

0

t

+ 1 2

a

t

2

v x



v

0

x

+

a x t x

 but look at components...

x

0 +

v

0

x t

+ 1 2

a x t

2

v y



v

0

y

+

a y t y



y

0 +

v

0

y t

+ 1 2

a y t

2 Phys 250 Ch3 p11

Example: A particle is confined to move in a horizontal plane. It starts at the origin at t=0. The particle has an initial velocity of 10 cm/s directed along the +x axis and an acceleration of 2 cm/s 2 in +y direction. Compute the particle’s position at t=1,2,3,4,5 s What is the particles velocity at t=1,3,5s?

Sketch the path of the particle by plotting the position at the indicated times.

Phys 250 Ch3 p12

Projectile Motion acceleration of gravity directed vertically down no horizontal acceleration (taking up as +y direction)

v x



v

0

x x



x

0 +

v

0

x t v y



v

0

y

-

gt y



y

0 +

v

0

y t

1 2

gt

2 Phys 250 Ch3 p13

Example: A ball is thrown horizontally from the leaning tower of Pisa with a velocity of 22 m/s. If the ball is thrown from a height of 49 m above the ground, where will the ball hit the ground?

discussion: monkey and the dart gun Phys 250 Ch3 p14

Projectile Range

y



y

0 +

v

0

y t

1 2

gt

2 

v

0

q

y

0 at impact 

t



R

2

v

0

y g R



x

-

x

0 

v

0

x t



v

0

x

2

v

0

y g



v

0 cos q 2

v

0 sin q

g

 2

v

0 2

g

sin q cos q

R

q 2 

v

0 2 sin 2 q

g

 90 q 1 (angles with the same range) R max 

v

0 2 ( at 45º )

g

Phys 250 Ch3 p15

Example: An arrow leaves a bow at 30 m/s.

What is its maximum range?

At what two angles could the archer point the arrow for a target 70 m away?

Phys 250 Ch3 p16