Transcript Trigonometry for Physics

Unit 4
Vectors & 2-D/Parabolic Motion
Physics Comp Book
p. 21
UNIT 4: Vectors & 2-D/Projectile Motion
Student will:
 SP1b. compare and contrast scalar and vector quantities;
 SP1c. compare algebraically the relationships between time,
distance, velocity and acceleration;
 SP1f. measure and calculate two dimensional motion
(projectile…) by using component vectors
Page
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Contents
parabolic motion, projectile
range, time of flight
trajectory, Pythagorean theorem
vectors, triangles and trigonometry diagrams
Ex: solving a 2-D problem
Lab SUMUP:
One dimension vs. two dimensions
 So far, we have only talked about the motion of
objects that travel in either the x or the y
direction… only one dimension at a time.
 What do these vectors look like:


a ball thrown directly up?
a car traveling on a highway?
One dimension vs. two dimensions
 …but sometimes, things don’t travel in a direction
that’s completely vertical or completely
horizontal…for example, what would these vectors
look like:
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A football field goal kick?
A basketball free throw shot?
How about a ballerina leaping across the stage?
A ball rolled off the edge of a table?
 Notice how the initial velocity is a certain direction
and the motion path doesn’t continue in that same
direction…look at the next examples, too…
In the absence of gravity a bullet
would follow a straight line forever.
With gravity it FALLS AWAY from
that straight line!
Football without gravity and with…
Basketball – with and without gravity
Hitting the target – aim high, not directly
at the target
BULLSEYE!
Projectile Examples
These are all examples of things that are
projected, then go off under the
influence of gravity

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Tennis ball
Golf ball
Football
Softball
Soccer ball
Bullet
 Hockey puck
 Basketball
 Volleyball
 Arrow
 Shot put
 Javelin
Not projectiles
 Jet plane
 Rocket
 Car (unless it looses contact with ground)
Why are these things NOT projectiles?
Height
Path of the Projectile
falling
rising
g
v
v
projectile
Distance downfield
(range)
Vertical
velocity
Horizontal velocity
 The key to understanding
projectile motion is to realize
that gravity acts vertically
 it affects only the vertical
part of the motion, not the
horizontal part of the motion
Vectors first, then the path…
 In order to describe the motion of objects like this,
we have to learn about the initial vectors FIRST.
 After that we will learn how to calculate:
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
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How high…
How far …
How long …
How fast…
Vectors at an angle…
 A vector in 2-D is made of an x-
y component
component and a y-component.
 Finding the components of a 2-D
vector is called RESOLVING the
vector
 To find the values of these
components, we use
trigonometry.
X component
Trigonometry Overview
opposite side
sin  
hypotenuse
adjacent side
cos  
hypotenuse
opposite side
tan  
adjacent side
opposite
y component
SOH CAH TOA

X component
adjacent
Example: Resolving a Vector
Write this example in your comp book p. 25
 A football is kicked with an initial velocity of
15 m/s at 30º above the horizon. What are
the x- and y-components?
Resolving a vector:
•Label parts of
triangle.
•Choose a trig
function for the
missing information.
•Use your calculator.

Vx = 15 cos 30 = 13 m/s
Vy = 15 sin 30 = 7.5 m/s
Using components to make a 2-D vector
 A vector in 2-D is made of an x-
y component
component and a y-component.
 When the vectors occur in ONE
dimension, you can just
add/subtract.
 But when you combine vectors
in 2-D, you have to use trig…
X component
Using components to make a 2-D vector
 Let the x- and y-components be the
sides of a right triangle.
 Draw a diagonal line. This is the
hypotenuse of the right triangle
y component
AND…
 It’s the RESULTANT vector.
 Use Pythagorean theorem to
determine the MAGNITUDE:
 a2+ b2 = c2
 Use tangent to determine the
DIRECTION (the angle).
X component
Example: Finding the resultant vector
Write this example in your comp book p. 25
 You walk 6 blocks east and then 13 blocks
north. What is your displacement from home?
Hint: solve for magnitude then direction…
13 blks
6 blks
Home
Example - Answer
a2  b2  c 2
13 blks
6 2  132  c 2
c 2  36  169
c  205
2
c  205
c  14.32 blocks from home
6 blks
Home
Example - Answer #1
opposite
sin(  ) 
hypotenuse
13 blks
 opposite 

  sin 
 hypotenuse 
1  13 
  sin 

 14.32 
  65.2 NofEast
1
14.32 blks

Home
6 blks
Demonstration
 We can see that the
horizontal and vertical
motions are independent
 The red ball falls vertically
 The yellow ball was given a
kick to the right.
 They track each other
vertically step for step and hit
the ground at the same time
Projectile motion – key points
The projectile has both a vertical and
horizontal component of velocity
2) The only force acting on the projectile once
it is shot is gravity (neglecting air
resistance)
3) At all times the acceleration of the projectile
is g = - 9.8 m/s2 (downward)
4) The horizontal velocity of the projectile does
not change throughout the path
1)
Key points, continued
On the rising portion of the path, gravity
causes the vertical component of velocity to
get smaller and smaller
6) At the very top of the path, the vertical
component of velocity is ZERO
7) On the falling portion of the path, the
vertical velocity increases
5)
More key points
If the projectile lands at the same elevation as
its starting point, it will have the same final
vertical SPEED as initial (opposite velocity)
9) The time it takes to get to the top of its path is
the same as the time to get from the top back to
the ground.
10) The range of the projectile (where it lands)
depends on its initial speed and angle of
elevation
8)
Example:
A 2.00 m tall basketball player wants to make a basket from
a distance of 10.0 m. If he shoots the ball at a 450 angle, at
what initial speed must he throw the ball so that it goes
through the hoop without striking the backboard?
y
y0
x
Maximum Range
 When an artillery shell is fired the initial speed of the
projectile depends on the explosive charge – this
cannot be changed
 The only control you have is over the angle of
elevation.
 You can control the range (where it lands) by
changing the angle of elevation
 To get maximum range set the angle to 45°
Interactive
 http://galileo.phys.virginia.edu/classes/109N/
more_stuff/Applets/ProjectileMotion/jarapplet.
html
 http://jersey.uoregon.edu/vlab/Cannon/
The ultimate projectile:
Putting an object into orbit
• Imagine trying to
throw a rock around
the world.
• If you give it a large
horizontal velocity,
it will go into orbit
around the earth!