1. Vectors and Two Dimensional Motion
Chapter 3

2. Constant V to the right, which path?

3. Addition of Vectors – Graphical Methods
One-Dimensional Vector Addition: If two vectors are in the same direction and along the same line, the resultant vector will be the arithmetic sum of the magnitudes and the direction will be in the direction of the original vectors. = =

4. Adding Vectors at some angle other than 0o or 180o:
R = A + B B A
Resultant (R) is drawn from the tail of the first vector to the tip of the last vector

5. Commutative Law of Addition
When two vectors are added, the sum is independent of the order of the addition.
A + B = B + A A B B R A

7. The vector that when added to A gives zero for the vector sum.
Negative of a Vector
The vector that when added to A gives zero for the vector sum.
A + (-A) = 0 A -A
A and –A have the same magnitude but point in opposite directions

8. We define the operation A – B as vector –B added to vector A.
Subtracting Vectors
We define the operation A – B as vector –B added to vector A.
A – B = A + (-B) B A
C = A - B -B

9. Multiplying a Vector by a Scalar
When vector A is multiplied by a positive scalar quantity m, then the product mA is a vector with the same direction of A and magnitude mA.
When vector A is multiplied by a negative scalar quantity -m, then the product -mA is a vector directed opposite A and magnitude mA.

10. Components of a Vector and Unit VectorsVy V q Vx
V = Vx + Vy
Vx = V cosq
Vy = V sinq

11. Y Vx V2
V = V1 + V2 Vy
V2y
V2x V1
V1y
V1x X
The components of V = V1 + V2 are:
Vx = V1x + V2x Vy = V1y + V2y

12. Two ways to specify a vector: We can give its components, Vx and Vy
We can give its magnitude V and the angle 
it makes with the positive x axis.
V2 = Vx2 + Vy2
tan  = Vy / Vx V
Vsinθ = Vy θ
Vcosθ = Vx

13. Analytically adding Vectors
Resolve each vector into its x and y components using sin and cos functions.
V1x = V1cos 
V1y = V1 sin 
Add the x components together and get the x component of the resultant
Add the y component together to get the y component of the resultant.
Vx = V1x + V2x
Vy = V1y + V2y

14. Find the magnitude and direction using the following:
V2 = Vx2 + Vy tan  = Vy / Vx

15. RELATIVE VELOCITY
When measuring the position or velocity, you must first define your frame of reference.
e.g. For a viewer on a ship or on the land. For one, the frame of reference is the ship, for the other, it is the land.

16. The first letter refers to the object
Example of notation:
VBS B = boat and S = shore
The first letter refers to the object
The second is the reference frame in which it has its velocity.
VBS = VBW + VWS
VWS
VBW
VBS

17. Done

18. Projectile Motion
Projectiles: Motion of objects given an initial velocity that then moves only under the force of gravity
Trajectory: The path a projectile follows
Motion of a projectile: Described in terms of its position, velocity, and acceleration.

21. Horizontal Motion Equations Vertical Motion Equations
vy = vyo + gt
vx = vxo
y – yo = vyot + 1/2gt2
x = xo + vxot
vy2 = vyo2 + 2g(y - yo)
Define initial direction of motion as positive.

22. Example 3
A ball rolls off a table 1.0 m high with a speed of 4 m/s. How far from the base of the table does it land ?

23. Analyzing Projectile Motionvy v vx
In projectile motion, the horizontal motion and the vertical motion are independent of each other. Neither motion affects the other.
X-Direction Constant Velocity
Y-Direction Constant Acceleration

25. Initial x and y Componentsvo
vyo q
vxo
vxo = vo cosq
vyo = vo sinq

26. Horizontal Motion Equations Vertical Motion Equations
vy = vyo + gt
vx = vxo
y – yo = vyot + 1/2gt2
x = xo + vxot
vy2 = vyo2 + 2g(y - yo)
Define initial direction of motion as positive.

27. R = Vo2 sin 2θ g

28. Maximum Height of a Projectile
vy = vyo - gt
0 = vo sinq - gt (at peak)
t =
vo sinq g
(at peak)
Dy = vyo t – ½ gt2 ( )
h = (vo sinq)
vo sinq g
- ½ g 2
h =
vo2 sin2q 2g

29. Horizontal Range of a Projectile
Dx = R = vxo t
R = vo cosq 2t (twice peak time)
t =
vo sinq g
(at peak)
R = vo cosq
2vo sinq g
sin 2q = 2sinqcosq (trig identity)
R =
vo2 sin 2q g

30. Example 4
An arrow is shot from a castle wall 10. m high. It leaves the bow with a speed of 40. m/s directed 37o above the horizontal.
a) Find the initial velocity components.
b) Find the maximum height of the arrow.
c) Where does the arrow land ?
d) How fast is the arrow moving just before impact ?

31. Example 5
A stone is thrown from the top of a building upward at an angle of 30o to the horizontal with an initial speed of 20.0 m/s.
a) If the building is 45.0 m high, how long does it take the stone to reach the ground ?
b) What is the speed of the stone just before it strikes the ground ?

32. Example 6 : 1985 #1
A projectile is launched from the top of a cliff above level ground. At launch the projectile is 35 m above the base of the cliff and has a velocity of 50 m/s at an angle of 37o with the horizontal. Air resistance is negligible. Consider the following two cases and use g = 10 m/s2, sin 37o = 0.60, and cos 37o = 0.80.
Case I : The projectile follows the path shown by the curved line in the following diagram.
a) Calculate the total time from launch until the projectile hits the ground at point C.
b) Calculate the horizontal distance R that the projectile travels before it hits the ground.
c) Calculate the speed of the projectile at points A, B, and C.

33. Example 7 : The Monkey Gun
Prove that the monkey will hit the dart if the monkey lets go of the branch (free-fall starting from rest) at the instant the dart leaves the gun.