given a right triangle with acute angle θ: θ
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Given a right triangle with acute angle θ: θTRANSCRIPT
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Warm-Up
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Projectile Motion
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Given a right triangle with acute angle θ:
Simple Trigonometry
adjacentoppositetan
hypotenuseadjacentcos
hypotenuseoppositesin
θ
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For the given triangle, calculate sine, cosine, and tangent:
You-Try #1
3.0
4.0θ
tancossin
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Make sure your calculator is set to degree mode (not radian mode)
Test: Type in sin(30) If you get 0.5, then you are in degree mode If you get anything else, then you are in a
different mode and need to switch to degree mode
Calculator - Degree Mode
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A component of a vector is its projection onto an axis (x-axis or y-axis).
Components of Vectors
sincosAAAA
y
x
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A small airplane leaves an airport on an overcast day and is later sighted 215 km away, in a direction making an angle of 22° east of north. How far east and north is the airplane from the airport when sighted?
Example 2
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A rocket is launched at an angle of 70.° above horizontal at 125 m/s.
How fast is it traveling horizontally? How fast is it traveling vertically?
Warm-Up: October 2, 2015You-Try #2
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x
y
yx
AA
AAA
1
22
tan
Combining Components
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A 45 kg box is moving on frictionless ice. Jon is pushing it east with a force of 75 N, while Roger is pushing it north with a force of 12 N.
What is the magnitude and direction of the net force on the box?
Example 3
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A car drives 25 miles east and 60. miles north.
What is the direction and magnitude of the car’s displacement?
You-Try #3
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A unit vector is a vector with magnitude of exactly 1.
The sole purpose of a unit vector is to specify a direction.
The unit vector in the positive x direction is The unit vector in the positive y direction is
Unit Vectors
ij
jAiAA yxˆˆ
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To add vectors: add the x-components add the y-components
Adding Vectors
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A baseball is hit at an angle of 37° above horizontal at 15 m/s.
How fast is it traveling horizontally? How fast is it traveling vertically?
Warm-Up: October 5, 2015
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An object, called a projectile is launched/thrown/shot at an initial velocity.◦ Initial velocity has a horizontal component and a
vertical component The only force on the object is gravity
◦ No thrust (object cannot be an airplane, rocket, etc.)◦ Drag is negligible
Its acceleration is the acceleration due to gravity◦ Gravity only pulls down (negative vertical direction)◦ The acceleration due to gravity is near the surface of
the earth The resulting motion is parabolic and is called
projectile motion.
Projectile Motion
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In projectile motion, the horizontal motion and the vertical motion are independent of each other.
Neither component’s motion affects the other.
Projectile Motion
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No horizontal acceleration
Horizontal velocity is constant
Horizontal motion is linear in time
Horizontal Projectile Motion
0xa
xx vv 0
tvxxtvxx x
000
00
cos
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Vertical acceleration due to gravity
Vertical velocity is linear in time
Vertical motion is quadratic in time
Vertical Projectile Motion
gay
gtvv
gtvv
y
yy
00
0
sin
2000
200
21sin
21
gttvyy
gttvyy y
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A baseball is hit from a height of 0.80 m above the ground with an initial velocity of 37 m/s at an angle 29° above horizontal.
a) What is the baseball’s maximum height?b) How far away does the baseball land?
Example 4
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A soccer ball is kicked so that its horizontal velocity is 12.6 m/s and its vertical velocity is 10.9 m/s. What is the magnitude and direction of the ball’s velocity?
Warm-Up: October 6, 2015
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80 Points possible Form B was significantly harder, so it was
graded out of 73. Common mistakes:
◦ Draw and label the resultant vector.◦ Acceleration doesn’t change; it’s ◦ Graphs
Average score was about 85%
Kinematics Test
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You will have an opportunity to retake the test. Before taking the retake, you must follow the test
corrections directions on the web site.◦ This is a LONG process.◦ Must be turned in by 3:30 this Friday.
Retake may only be taken:◦ Monday, October 12 at lunch◦ Tuesday, October 13 at lunch◦ Tuesday, October 13 after school
Retake will be more difficult that the original test. The highest grade you can get is 85%. If you get lower than your original grade, then
you keep your original grade.
Test Retake
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Test Answers – page 4
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Other Test Questions?
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A bullet is shot from the ground at 485 m/s at an angle 70.0° above the horizontal. How far away will it land?
You-Try #4
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A ball rolls off of a 120 cm tall horizontal table at 7.25 cm/s. How far away from the table does the ball land?
Warm-Up: October 7/8, 2015
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On the website Some of the problems are difficult Some of the problems are very difficult
#1-2 due Monday #3-4 due Tuesday #5-6 due Thursday #7-8 due Friday
If you have time after today’s lab, work on the worksheet with your partner
Projectile Motion Worksheet
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Groups of 3-4 (assigned by Mr. Szwast)
Measure and record the height of the table. Follow procedures #1-8 (including the
graph)◦ Your graph should fill up a full page of graph
paper Answer #1-2 before attempting extra credit. You may answer #3 after attempting extra
credit.◦ You only get one chance at the extra credit.
Projectile Motion Lab
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A baseball is thrown from a height of 1.87 m at 27.3 m/s, 15° above horizontal, to the first baseman, 9.5 m away. Which of the following will happen?
a) The ball will go over the first baseman’s head.
b) The first baseman will catch the ball before the ball has a chance to hit the ground.
c) The ball will hit the ground before it makes it to the first baseman.
Warm-Up: October 9, 2015
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On the website Some of the problems are difficult Some of the problems are very difficult
#1-2 due Monday #3-4 due Tuesday #5-6 due Thursday #7-8 due Friday
If you have time after today’s lab, work on the worksheet with your partner
Projectile Motion Worksheet