Chapter 3 Motion in Two Dimensions (2D)

Vectors and Projectile Motion

x

y

What is a …?

• Vector– Has Magnitude & Direction

12 m/s Northeast– Examples:

• Drawn as arrows• Angles measured from 0

• Scalar– No direction– Examples:

12 m/s

tip

tail

0°

+

Vector addition• Vectors can be added together 2 ways:

graphically or analytically (with math!)• The answer to a vector addition problem is

called the Resultant, which is also a vector, so it has a magnitude (size) and direction

• The Resultant (R) goes from the starting tail to the ending tip

• Any number of vectors can be added with these methods

• http://illuminations.nctm.org/Activity.aspx?id=6598

v1

v2R

Graphical Addition

• Add using “Tip to Tail” method—tail of the next vector is drawn on the tip of the previous

• Draw to scale, keeping size and direction. Use ruler & protractor.

• Start at an origin!• http://lectureonline.cl.msu.edu/~mmp/kap3/cd052a.htm

• Examples next

Vector Direction: ExamplesWrite the vector angle in terms of 0 to 360°

A. 50m North

B. 100m West

C. 50m South

D. 100m East

0°Now add (sketch) A+D and B+C

AB

C

D

http://phet.colorado.edu/simulations/sims.php?sim=Vector_Addition

Vector addition

• Properties of vectors:– Vectors can be moved parallel to themselves

in a diagram– Vectors can be added in any order– To subtract a vector, add the vector pointing

in the opposite direction (or negative)– Multiplying or dividing vectors by scalars

results in vectors– http://www.nctm.org/standards/content.aspx?id=26779

– http://illuminations.nctm.org/Activity.aspx?id=6598 storm

Analytical AdditionWe’ll only look at vectors which are perpendicular to each other, so Right Triangles!

•Use Pythagorean theorem and/or Trigonometry to find exact values for Resultant SOH CAH TOA

– One axis is x, the other is y—they are independent of each other

– Make a sketch to help—add “tip to tail”, draw “x” first– Always indicate from 0, use sketch to reduce errors

•Carefully determine the correct quadrant for Resultant, R

Example: Perpendicular vectors Little red riding hood walks to grandma’s house 165 m North and 125 m to the East. What is her resultant displacement?

Example in packet

R = 207 m

E

N

R

tan = y/x = 165m/125m

= tan-1 (1.32) = 52.9°

Next…

• When you know the vector, but want the x and y that make it up:

• Breaking the vectors into its pieces is called Resolving the vector

• Use Trig to get the horizontal and vertical parts, or components, of the original vector

Resolving Vectors into Components

sin

siny

yAop

A

p

hyp A

A

cos

cosx

xAad

A

j

hyp A

A

Vector A and its x and y components

Often vector length and angle are known

FORMULAS FOR FROM X AXIS ONLY

Example What are the components that make up the given vector V?

V = 10 m/sec

60°

Vx = Vcos = 10cos60° = 5 m/s

Vy = Vsin = 10sin60° = 8.66 m/s

Examples

cos = Vx / V

sin = Vy / V

What is a Projectile?

Projectile motion• Projectile: Moving object that is only

affected by gravity.• The motion of a projectile is determined

only by the object’s initial velocity and gravity (g). vi has vy and vx components.

• Use x & y dimensions to analyze motion.• Vertical motion (y) of a projected object is

independent of its horizontal motion. • Vertical motion of a projectile is nothing

more than free fall (acceleration).• Horizontal motion (x) is constant

velocity.• The common variable between the

horizontal and vertical motion is time

Path of a Projectile• The path of a projectile is called its

trajectory.• The trajectory of a projectile in free fall is a

parabola. • http://regentsprep.org/Regents/physics/phys06/amotproj/sld005.htm

i

Variables:

∆xvx

∆t∆yviy

vfy

ay

vi and so also:vx= vicosviy= visin

Projectile Motion Problems

• The first type of projectile motion is: Horizontal Launch Projectiles – Path appears to be ½ a parabola

– Projectile has no initial vertical velocity, viy = 0

– Vertical is done like a dropped object (free fall)

– Projectile has no horizontal acceleration, ax = 0

– Horizontal motion portion is done the same as a constant velocity object (vix = vfx = vx)

– Use previous equations with revised subscripts

Question ?

• If a gun is fired horizontally and simultaneously a bullet is dropped from the same height, which bullet hits the ground first?

Variables:

∆xvx

∆t∆yViy = 0

vfy

Ay

Angle = 0

Projectile formulasx-direction y-direction

cos sin

x y

x

v v v v

xv

t

2 2

2

2

1

21

2

fy iy y

fy iy y

fy iy

iy y

v v a t

v v a y

y v v t

y v t a t

** Note:

vi is the vector sum (resultant) of vx and viy

vf is the vector sum of vx and vfy

Constant v

Free fall

Example A ball is launched horizontally at 10 m/s off

of a 1.2 m tall table. Calculate the following:

A) The time in the air.B) How far from the table does the ball land

(x)?C) What is the magnitude of the velocity of

the ball just as it hits the ground?

Examples in packet

Other horizontal examples

• Drop a bomb

• Shoot

• Throw or drive off a cliff…..

• All are initially going horizontally only, the x direction (an angle of 0 or 180)

• True Lies…

• two more questions…

..\My Videos\robbiemadison motorcycle.mp4

Horizontal Projectile

• An airplane flying steadily and horizontally drops a bowling ball. If it is in free fall, where does it hit relative to the plane?

• In front, directly under, or behind?

Horizontal Projectile

• What would happen if you shoot a projectile from a moving vehicle and the projectile has the same speed as the vehicle, but opposite direction?

• https://www.youtube.com/watch?v=BLuI118nhzc

Projectile Motion Problems• The second type of projectile motion is for objects

Launched at an Angle (Vi and )• These objects will follow a full parabola• The up ½ of the motion = the down ½ at

corresponding heights, other than direction (it is symmetric)

• Initial velocity is a vector at an angle which must be resolved into x & y components

• Same equations work as for Horizontal projectiles, but viy is not = 0.

• Use vx = vi·cos, viy = vi·sin..\My Videos\robbiemadison motorcycle.mp4

y=0 Projectile Motion Problems

http://jersey.uoregon.edu/newCannon/nc4.html

http://jersey.uoregon.edu/newCannon/nc1.html

http://www.mhhe.com/physsci/physical/giambattista/proj/projectile.html

Total time in the air (ttotal) = time to go up + time to go down

vi = initial velocity (total), includes both vx and viy

vf = final velocity (total); = magnitude of vi (or vx + vfy )

ymax = maximum height (top). Note: vy = 0

xmax = total horizontal distance, called Range (R)

Range symmetry and maximum

http://phet.colorado.edu/simulations/projectilemotion/projectile.swf

Problem Solving

• Angled launch problems are solved similar to horizontal

• Same procedure except for the start: Resolve velocity vector into x & y components (viy is not zero)

• Use symmetry and Vy = 0 at top when needed

• examples

Example A cannon ball gets fired at a 30 degree angle at 26 m/sec.A) How far away does it land?B) How long is it in the air?C) What is the maximum height?D) How fast is it moving when it hits the ground?E) How fast is the ball moving and how high is the

ball 0.6 sec after launch?

http://jersey.uoregon.edu/newCannon/nc4.html

http://jersey.uoregon.edu/newCannon/nc1.html

More practice:

• Practice Problem 1:• A golf ball is hit with a velocity of 31 m/s at 48

above the horizontal. – (A) Find how long it is in the air. – (B) How high did the ball go? – (C) What was the range of the ball?

• Challenge Problem:• A cannon shoots a pumpkin at an angle of 30

degrees which lands 120 m from the cannon. – A) With what velocity was the projectile fired? – B) What is the pumpkin’s velocity as it strikes the

ground?– https://www.youtube.com/watch?v=b0xmTzSXL0E

Monkey Hunter!!

http://www.phy.ntnu.edu.tw/ntnujava/index.php?PHPSESSID=04e5251cd5a765c0ee22dd61a2040312&topic=144.msg721#msg721

http://www.physics.umn.edu/outreach/pforce/circus/

http://www.mhhe.com/physsci/physical/giambattista/proj/projectile.html http://

www.waowen.screaming.net/revision/force&motion/

mandh.htm