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2-D Kinematics
Definition: Vector - a quantity that has both magnitude & direction
-ex. velocity, acceleration, displacement, force
Definition: Scalar - a quantity that has only magnitude
-ex. speed, displacement, mass, time
-vector quantity can be represented graphically using rays (arrows):
-in 1 dimension...vectors can be added by placing the tail of the first vector
at the head of the other
Ex.
:
Definition: Resultant - a 3rd vector which represents the sum of vectors
-i.e. 20 m/s and 10 m/s vectors as shown above
-order of addition does not matter
-always use compass rose to indicate direction
Ex. You travel 50 m north. Then you travel 40 m east.
What is your resultant displacement ?
In 2 dimensions...{using graph paper}...
Resultant vector...
50 2  40 2  64.03 m
40m
50m
 40m 
0
  tan 1 
  38.7
50
m


tan  
So final answer is…
64.0 m [N38.7oE]
Note: total distance travelled = 50 m + 40 m = 90 m (scalar)
1. Choose a scale, i.e. 1 cm = 10 m
2. Draw vectors to scale, tail to tail at the appropriate angle
3. Move first vector's tail to head of second vector, keeping the same angle
4. Join tail of second vector to head of first with a dotted line
5. Measure length and direction of vector or use Pythagorean Theorem or Sine Law or Cosine Law
Sine Law
Cosine Law
Addition of Several Vectors:
-same as addition of 2 vectors
-order of addition not important
Ex.: An explorer walks 13 km due east, then 18 km due north, and finally 3 km west.
a. What is the total distance travelled by the explorer? (34 m)
b. What is the resulting displacement from the starting point?
Analytical Method of Vector Addition:
-sum of any 2 vectors can be determined using
TRIG: SOH CAH TOA
-effective for adding perpendicular vectors where right angles are formed
-where non-right triangles are formed, use Sine or Cosine Laws
Vector Components:
-consider a single vector (i.e. resultant) being broken down into 2 component vectors, perpendicular to each other
-this is called vector resolution
-used in projectile motion
Subtraction of Vectors:
-done like addition of vectors
 
A B 
 
A A 


A  ( B)


A  ( A)  0
Definition: Relative Velocity – the velocity of an object that appears to have an observer who is moving with a different
object
-calculated by determining difference (subtraction) of two vectors
3 types of relative motion problems:
1.
2.
3.
Calculation of velocity of a moving object as perceived by an observer who may or may not be moving with
respect to some arbitrary frame of reference (such as the ground).
“River problems” – determine the motion of a boat disturbed by the motion of water flowing around it (current) as it
crosses a river
-usually involves right angle triangles  use Pythagorean Theorem
“Airplane problems” – similar to river problems
-usually involve non-right triangles  airplane is affected by wind around it
-wind can be at any angle relative to airplane
-use Law of Sines/Cosines
Projectile Motion
Definition: Projectile -a moving object given initial velocity that then moves only under the force of gravity
-the path followed by a projectile is called a trajectory (parabolic)
-shape of trajectory and horizontal motion depend on frame of reference of the observer
-vertical motion does not depend on frame of reference of observer
-vertical and horizontal motion of a projectile are independent of each other!
Neglecting air resistance... let x be horizontal displacement, let vx be initial horizontal velocity, & let t be time
x  vx  t
{remember... v x final  v x initial }
For falling objects, g = -9.81 m/s2…let y be vertical displacement, let vy be initial vertical velocity, let t be time
1 2
gt
2
 v y  gt
y  vyt 
v y final
Example:
A stone is thrown horizontally at 15 m/s from the top of a cliff 44 m high.
a. How long does the stone take to reach the bottom of the cliff?
b. How far from the base of the cliff does the stone strike the ground?
c. Sketch the trajectory of the stone.
Objects Launched at an Angle
-horizontal component, vx, is constant
-since gravity acts on the projectile...vy becomes less as the ball reaches the top of the trajectory
-at the top of trajectory, vy = 0 m/s and vx is still constant
-after the top of trajectory, vy increases and becomes -ve (downward)
-vertical speed, vx, is the same throughout
Definition: Range (R) -horizontal distance from the point of bounce until projectile returns to surface height
Problem-Solving Tips for Objects Launched at an Angle
1. Determine vx and vy components of initial velocity.
2. Solve parts of the problem involving each component separately.
3. Use symmetry of the trajectory (parabola).
a) Launching and landing heights are equal
b) Rising and falling times are equal
c) Horizontal distances moved in each half of trajectory are equal
4. Remember, ignore air resistance!
Any velocity (or any other vector quantity!) can be resolved or "broken down" into a horizontal (vx) component and a
vertical (vy) component ---> vector resolution
Vector resolution is done by using trigonometry! YEAH!
v x  v  cos 
v
v y  v  sin
vy
vx
-when the vx and vy are connected "head-to-tail" ---> a right triangle is formed
-the hypotenuse of the triangle is initial velocity v
Example:
A football player kicks a football from ground level with a velocity of magnitude 35 m/s at an angle of 30 ° above the
horizontal. Find:
a. its "hang time" or the time in the air,
b. the distance the ball travels before it hits the ground
c. its maximum height.
Repeat this problem using angles of 45 ° and 60 ° above the horizontal.
Formulas for Projectile Motion:
Horizontally…
Vertically…
x  vx  t
v x  v  cos 
1 2
gt
2
 v y  gt
y  vyt 
v y final