Vectors and 2D Motion

¥ Vectors and Scalars ¥ Vector arithmetic ¥ Vector description of 2D motion ¥ Projectile Motion ¥ Relative Motion -- Reference Frames

Vectors and Scalars

Scalar quantities: require magnitude & unit for complete description

Examples: mass, tim e, temperature, speed, …… (what others?)

2.7 kg 57 ¡C 60 m/s

Vector quantities: require magnitude, unit & direction for complete description

Examples: displacement, velocity, acceleration …… (what others?)

500 m north 50 m/s heading 040¡ 9.8 m/s2 down

Vector Notation

Vector quantities are graphically represented as arrows …….

y A The length of the arrow represents the magnitude of the vector and the direction is self-evident. θ x

Vectors quantities are referred to by symbol, such as… “A” or “A” (arrow used on blackboard, boldface in the text, and on overheads)

A simple, unbold, unarrowed “A” refers to the magnitude of vector A.

A = |A|

1 Vector Math

Vectors can be multiplied by scalars:

-B 3B B

0.75B

The result is that the magnitude of the vector changes, but not the direction, except in the case of muliplication by a negative number where the vector reverses direction.

Adding Vectors (parallelogram method)

R

A B

A + B = R

Adding Vectors ( “tip-to-tail” method)

R B

A

A + B = R

2 Adding Vectors ( “tip-to-tail” method)

R The result is A independent of the order of addition! B + A = A + B B

B + A = R

Subtracting Vectors B -B

A A R R

A + B = R A = R - B

R A

B

Vector Components y A

Ay θ x Ax Vector A (with magnitude, A) is directed at an angle θ above the +x axis

A = Ax + Ay

Ax: vector component of A in the x-direction

Ay : vector component of A in the y-direction

3 Scalar Components y A

Ay θ x Ax Given A, θ, the scalar components of A are

θ) Ax = A cos ( θ) Ay = A sin (

Scalar Components y Given Ax, Ay, how do you find A, θ ? (-,+ ) (+ ,+ )

x Magnitude: (-,-) (+,-) =+22 AAAxy Angle: Be careful! ¥Calculator in degree mode? A ¥Look at the signs of A and A θ = −1 y x y. tan ( ) Does the angle make sense? Ax ¥Inverse tangent only gives back a result f rom -90¡ to +90¡. How do you get the right quadrant?

Adding Vectors by Components +y +y +y

C B By C By C C A y Bx θ Ay Ay A A B x +x x x +x Cx +x A+B=C

1. Choose coordinate system and draw a picture.

2. Find scalar components: Ax, Ay, Bx, By 3. Calculate scalar components:

Cx =Ax+ Bx and Cy =Ay+ By C =+22 θ = −1 y 4. Find:CCCxy 5. Find: tan ( ) Cx

4 Example N Adding Displacement Vectors:

A hikers walks D3 D1 = 6.0 km east on day 1 30¡ D D2 = 4.0 km north on day 2 T θ D2 D3 = 10.0 km at 30.0¡ north of west on day 3. W E D Find her total displacement for the 1 Trip. By graphical Methods

DT = D1 + D1 + D1 S

Example N Analysis

x-component y-component D3 30¡ D1 +6.0 km 0 km DT θ D2 D2 0 km +4.0 km W E

D3 -10 cos(30¡) +10 sin(30¡) D1 = -8.7 km = +5.0 km By graphical Methods DT -2.7 km +9.0 km

Dkmkm=−(.)(.)2722 + 90 = 94 . T S

− 90. θ =+180ooootan1 ( ) =−=180 73 107 (Needed to add 180¡ to get result in − 27. the correct quadrant)

Vector Description of Motion

Average Vector Displacement: ∆rr=− r f i Average Vector Velocity: ∆r ∆v = ∆ t Instantaneous Vector Velocity: A particle moves from P (at ti)to Q(at tf) along the trajectory shown above. ∆r ∆v = lim (A trajectory is a path of motion ∆t→0 ∆t through space)

5 Vector Description of Motion Velocity vectors How does velocity change? shown in red vf

vi ∆v

v ∆vv=− v f f i vi Average Vector Acceleration: ∆v ∆a = The velocity vector’s direction is ∆t always tangent to the trajectory and in the direction of the motion. Instantaneous Vector Acceleration:

The length of the velocity vector shows ∆v the instantaneous speed of the particle. ∆a = lim ∆t→0 ∆ Longer means faster. t

Vector Description of Motion Velocity vectors shown How does velocity change? in red, acceleration vectors in purple vf ai vi ∆v af v ∆vv=− v f f i vi Average Vector Acceleration: ∆v ∆a = The velocity vector’s direction is ∆t always tangent to the trajectory and in the direction of the motion. Instantaneous Vector Acceleration:

The length of the velocity vector shows ∆v the instantaneous speed of the particle. ∆a = lim ∆t→0 ∆ Longer means faster. t

Vector Description of Motion

As a particle moves along its trajectory,

Acceleration vector shows how the velocity vector changes.

Velocity vector shows how the position changes

Both velocity and acceleration can change in both magnitude and direction

(This is a quicktime animation in the screen version )

6 Projectile Motion

Projectile Motion describes the motion of any object thrown into the air at any arbitrary angle.

To describe projectile motion mathematically, we assume:

1. Projectile motion is uniform ly accelerated motion. The acceleration vector is directed Quicktime Movie of a juggler vertically downward and has a magnitude of g=9.8 m/s2 .

2. The effects or air resistance are negligible.

Analyzing Projectile Motion

+y The projectile is launched at θ initial velocity, v0 at angle 0 above the horizontal. v0 (x,y) v Initial components are: θ vv= cosθ 0 +x x00 0 = θ (x0,y0) vvy00sin 0 Subsequent to launch, the x-component of velocity, remains constant; only the y-component changes.

x - motion y - motion ==θ = vv=− gt vvxx00 vcos 0constant yy0 xvtv==( cosθ ) t −= −1 2 x000 yy00 vty 2 gt 2 =−2 − vvyy0 2( gyy0 ) The speed at time t is given by: =+22 vvvxy

Example:

Find components first!

==o vmsmsx0 30cos 30 / 26 / vmsms==30sin 30o / 15 / y0

A ball is thrown off a 25 m high roof at a speed of 30 m/s at an angle of 30¡ above horizontal. 2 =−⇒=+2 1) What is the maximum height? vgyyyyvgyy0 22()000 / y=+25 m(/)/(./) 15 ms22 19 6 ms = 36 m

−= − 2 2) At what time does it land? yy00 vty 05. gt 4..,. 9tt2 −−=⇒=− 15 25 0 t 1 2 sts =+ 4 3 3) How far does it travel? (What is the horizontal range?) −= xx00 vtx xm=+0()(.) 26 4 3 = 110

7 Relative Motion

How is a motion described by differerent observers in different reference frames which are in motion with respect to each other?

Relative Position Relative Velocity

Relative Velocities

Consider the motion of a boat in a river.

vbr : velocity of boat seen by river vre : velocity of river seen by earth vbe : velocity of boat seen by earth

Vector equation:

vbe = vbr + vre

Sign Convention:

vre = - ver

Example

At what angle do you point the boat to go straight across the river?

vbr : speed is 20 m/s, direction ??? vre : current, 10 m/s, due east vbe : speed is ???, due north N Vector equation:

vbe = vbr + vre In components: E

East: (vbr)x + (vre)x = (vbe)x -20 m/s sin(θ)+10 m/s = 0 ---> θ =30¡

North: (vbr)y + (vre)y = (vbe)y 30° 20 m/s cos( )+0 m/s = (vbe)y = vbe = 17 m/s

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