Physics 121

3. Kinematics in 2-D and Vectors

3.1 Vectors and Scalars

3.2 Addition of Vectors - Graphically

3.3 Subtraction and Scalar Multiplication

3.4 Addition of Vectors - Analytically

3.5 Projectile Motion

3.6 Projectile Motion Problems

3.7 Projectile Motion is Parabolic

3.8 Relative Velocity

Example 3.1 . . . Can 3 + 4 = 5?

Give examples of Scalar and Vector quantities

You drive 3 miles East and then 4 miles North. How far are you from your starting point? 5 miles

What is the difference between distance and displacement?

Distance is a Scalar (has magnitude only) and displacement is a Vector (direction is also required).

Scalars: speed and densityVectors: velocity and force

Trig Review

Pythagorean Theorem: a2 + b2 = c2

sin = opposite / hypotenuse = a /c

cos = adjacent / hypotenuse = b / c

tan = opposite / adjacent = a / b

c

b

a

Example 3.2 . . . Vector Addition

You drive 3 miles East and then 4 miles Northeast. What is your net displacement (magnitude and direction)?Solve it (a) graphically (b) analytically

Solution 3.2(a) . . . Graphical Vector Addition

3

4

450

?

?

E

N

Solution 3.2(a) . . . Graphical Vector Addition

3

4

450

6. 4

260

E

N

Resultant displacement vector is 6.4 m, 260 North of East

Solution 3.2(b) . . . Analytical Vector Addition

Vector X = V cos Y = V sin

3 at 00 3 0

4 at 450 2.8 2.8

Resultant 5.8 2.8

R2 = X2 + Y2

R2 = (5.8)2 + (2.8)2

R = 6.4

tan = Y / X

tan = 2.8 / 5.8

= 260

A picture is worth 103 words !!!

3

4

2.83

2.8

5.8

2.86.4

Projectile Motion

1. Which projectile reaches the ground first?

2. Which is moving faster just before impact?

Two objects are launched horizontally with different speeds

Projectile Motion

1. Same Time

2. Different Speeds

Two objects are launched horizontally with different speeds

Example 3.3 . . . Projectile Motion

Given h= 20 m and VH = 7 m/s. Calculate (a) Time to hit the ground (b) Range D (c) Impact Speed

h

D

Solution 3.3 . . . Projectile Motion

V2 = (20)2 + (7)2

V = 21.2 m/s

h = ½ a t2

t = 2 s

D = (7 m/s)(2 s)

D = 14 m

Vv = 0 + atVv = 0 + (10)(2)Vv = 20 m/s

Example 3.4 . . . Soccer Ball

A soccer ball is kicked and it is projected with a speed of 20 m/s at an angle of 300. How far away does it land?

D

Solution 3.4 . . . Soccer Ball

D = (VH )(t) D = (20 cos 300)(2)D = 34.6 m

Vv = 20 sin 300 = 10 m/sSo air time is 1s (up) and 1s(down) = 2 s

That’s all folks!