Unit 4Vectors & 2-D/Parabolic Motion

Physics Comp BookUNIT 4: Vectors & 2-D/Projectile MotionStudent 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 Contents22 parabolic motion, projectile23 range, time of flight 24 trajectory, Pythagorean theorem 25 vectors, triangles and trigonometry diagrams26 Ex: solving a 2-D problem27 Lab SUMUP:

p. 21

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: 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 fromthat 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

Tennis ball Golf ball Football Softball Soccer ball Bullet

Hockey puck Basketball Volleyball Arrow Shot put Javelin

These are all examples of things that are projected, then go off under the

influence of gravity

Not projectiles Jet plane Rocket Car (unless it looses contact with ground)

Why are these things NOT projectiles?

Path of the Projectile

v

Distance downfield(range)

Hei

ght rising falling

projectile

g

Horizontal velocity

Verticalvelocity

v

The key to understandingprojectile motion is to realize that gravity acts vertically

it affects only the vertical part of the motion, not thehorizontal 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: How high… How far … How long … How fast…

Vectors at an angle… A vector in 2-D is made of an x-

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.

Vecto

r

X component

y co

mpo

nent

Trigonometry Overview

sin

cos

tan

opposite side

hypotenuse

adjacent side

hypotenuse

opposite side

adjacent side

SOH CAH TOA

Vecto

r

X component

y co

mpo

nent

Hypot

enus

e

op

po

site

adjacent

Example: Resolving a Vector

A football is kicked with an initial velocity of 15 m/s at 30º above the horizon. What are the x- and y-components?

Vx = 15 cos 30 = 13 m/s

Vy = 15 sin 30 = 7.5 m/s

Resolving a vector:

•Label parts of triangle.

•Choose a trig function for the missing information.

•Use your calculator.

Write this example in your comp book p. 25

Using components to make a 2-D vector

A vector in 2-D is made of an x-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…

Vecto

r

X component

y co

mpo

nent

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 AND…

It’s the RESULTANT vector. Use Pythagorean theorem to

determine the MAGNITUDE: a2+ b2 = c2

Use tangent to determine the DIRECTION (the angle).

Vecto

r

X component

y co

mpo

nent

Example: Finding the resultant vector

You walk 6 blocks east and then 13 blocks north. What is your displacement from home?

Hint: solve for magnitude then direction…

Home

6 blks

13

blks

Write this example in your comp book p. 25

Example - Answer

Home

6 blks

13

blks

a2 b2 c 2

62 132 c 2

c 2 36 169

c 2 205

c 205

c 14.32 blocks from home

Example - Answer #1

Home

6 blks

13

blks

14.32 blks

NofEast

hypotenuse

opposite

hypotenuse

opposite

2.65

32.14

13sin

sin

)sin(

1

1

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

1) 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

Key points, continued

5) 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

More key points

8) 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

Example:

A 2.00 m tall basketball player wants to make a basket froma distance of 10.0 m. If he shoots the ball at a 450 angle, atwhat initial speed must he throw the ball so that it goes through the hoop without striking the backboard?

y

xy0

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 aroundthe world.

• If you give it a large horizontal velocity,it will go into orbitaround the earth!