Given a right triangle with acute angle θ: θ.

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Warm-Up

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 Given a right triangle with acute angle θ: θ

Transcript of Given a right triangle with acute angle θ: θ.

Page 1: Given a right triangle with acute angle θ: θ.

Warm-Up

Page 2: Given a right triangle with acute angle θ: θ.

Projectile Motion

Page 3: Given a right triangle with acute angle θ: θ.

Given a right triangle with acute angle θ:

Simple Trigonometry

adjacentoppositetan

hypotenuseadjacentcos

hypotenuseoppositesin

θ

Page 4: Given a right triangle with acute angle θ: θ.

For the given triangle, calculate sine, cosine, and tangent:

You-Try #1

3.0

4.0θ

tancossin

Page 5: Given a right triangle with acute angle θ: θ.
Page 6: Given a right triangle with acute angle θ: θ.

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

Page 7: Given a right triangle with acute angle θ: θ.

A component of a vector is its projection onto an axis (x-axis or y-axis).

Components of Vectors

sincosAAAA

y

x

Page 8: Given a right triangle with acute angle θ: θ.

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

Page 9: Given a right triangle with acute angle θ: θ.
Page 10: Given a right triangle with acute angle θ: θ.

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

Page 11: Given a right triangle with acute angle θ: θ.

x

y

yx

AA

AAA

1

22

tan

Combining Components

Page 12: Given a right triangle with acute angle θ: θ.

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

Page 13: Given a right triangle with acute angle θ: θ.

A car drives 25 miles east and 60. miles north.

What is the direction and magnitude of the car’s displacement?

You-Try #3

Page 14: Given a right triangle with acute angle θ: θ.

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

Page 15: Given a right triangle with acute angle θ: θ.

To add vectors: add the x-components add the y-components

Adding Vectors

Page 16: Given a right triangle with acute angle θ: θ.

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

Page 17: Given a right triangle with acute angle θ: θ.

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

Page 18: Given a right triangle with acute angle θ: θ.

In projectile motion, the horizontal motion and the vertical motion are independent of each other.

Neither component’s motion affects the other.

Projectile Motion

Page 19: Given a right triangle with acute angle θ: θ.

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

Page 20: Given a right triangle with acute angle θ: θ.

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

Page 21: Given a right triangle with acute angle θ: θ.

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

Page 22: Given a right triangle with acute angle θ: θ.
Page 23: Given a right triangle with acute angle θ: θ.

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

Page 24: Given a right triangle with acute angle θ: θ.

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

Page 25: Given a right triangle with acute angle θ: θ.

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

Page 26: Given a right triangle with acute angle θ: θ.

Test Answers – page 4

Page 27: Given a right triangle with acute angle θ: θ.

Other Test Questions?

Page 28: Given a right triangle with acute angle θ: θ.

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

Page 29: Given a right triangle with acute angle θ: θ.
Page 30: Given a right triangle with acute angle θ: θ.

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

Page 31: Given a right triangle with acute angle θ: θ.

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

Page 32: Given a right triangle with acute angle θ: θ.

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

Page 33: Given a right triangle with acute angle θ: θ.

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

Page 34: Given a right triangle with acute angle θ: θ.

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