given a right triangle with acute angle θ: θ

Warm-Up

Projectile Motion

Given a right triangle with acute angle θ:

Simple Trigonometry

adjacentoppositetan

hypotenuseadjacentcos

hypotenuseoppositesin

θ

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

You-Try #1

3.0

4.0θ

tancossin

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

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

Components of Vectors

sincosAAAA

y

x

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

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

x

y

yx

AA

AAA

1

22

tan

Combining Components

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

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

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

You-Try #3

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

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

Adding Vectors

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

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

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

Neither component’s motion affects the other.

Projectile Motion

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

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

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

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

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

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

Test Answers – page 4

Other Test Questions?

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

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

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

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

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

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