Motion in Two DimensionsVectors and Projectile Motion

WHS AP Physics

What we need to know…

Add, subtract, and resolve displacement and velocity vectors, so we can: determine components of a vector along two perpendicular axes determine the displacement and location of a particle relative to

another determine the change in velocity of a particle or the velocity of

one particle relative to another Understand the motion of projectiles in a uniform

gravitational field, so we can: Determine the horizontal and vertical components of velocity and

position as functions of time Analyze the motion of a projectile that is projected with an

arbitrary initial velocity

What is a vector?

Objects do not always move in a straight line To be accurate in

analysis, you need two things… the value measured and direction

A vectors are physical quantities that have both a magnitude and a direction

What is a vector?

Magnitude is the scalar part of a vectorScalar – A physical

quantity that has no direction… just a number and units

Distance and Speed are scalars “3 km”, “30 m/s”

Displacement, velocity, and acceleration are vectors “3km north”, “30 m/s at 60 degrees”, a = 5i + 4j + 3k

Symbology

In the book Scalars are in italics…. v Vectors are in boldface with an arrow over the

variable….

On the board and in your work Vectors have an arrow over the variable symbol…

In PPT, bold or with an arrow

v

Vectors

Graphically Vectors are depicted as arrows Length is relative to magnitude Arrowhead indicates direction

Since Vectors depict a magnitude and direction Vectors can be moved graphically as long as the magnitude and

direction are unchanged

a

Graphic Vector Addition

Triangle Method Tip to Tail

Resultant A vector representing the sum of two or more vectors

ab

R=a+b

b

Graphic Vector Addition

Parallelogram Method Tail to Tail

Vectors can be added in any ordera+b = b+a

ba

b

R=a+b

Adding Vectors

Vectors add in any order

Vector Subtraction

A – B = A + (-B)-B has equal magnitude but has the opposite direction

Does A – B = B – A?

A-B

B

-B

A-B

Scalar Multiplication

AB Resultant is a vector in the same direction Magnitude is A times the magnitude of B

Example: 3B=?

B B B

3B

Try this…

An A-10 normally flying at 80 km/hr encounters wind at a right angle to its forward motion (a crosswind). Will the plane be flying faster or slower than 80 km/hr?

60 km/hr

Crosswind

80 km/hr

Is the resultant greater than or less than 80 km/hr?

80 km/hr

60 km/hr

Add the vectors

Resultant

hr

km100

6080Resultant 22

2-D Coordinate System

To analyze motion, we need a frame of reference (FOR) In 2-D, a good reference is

the x-y coordinate plane In our FOR, the vector has a

magnitude, r, and a direction angle, qX

Y

Remember… we can move a vector as long as we don’t change magnitude or direction

r

q

Analyzing vectors

X

Y

Once you establish a coordinate system or frame of reference, you can begin to analyze vectors mathematically

First step… resolve the vector into its x-component and its y-component

The component vectors can represent the change in x and the change in y for the vector

r

q

Dx

Dy

Resolving Components

If we know the magnitude, d, and direction, ,q we can find the x and y components using Trigonometry

X

Y

Dx

Dyr

q

cos

sin

,

cossin

rx

ry

orr

xand

r

y

Unit Vectors

Any vector can be represented as the vector sum of its components

“Unit vectors” are used to specify the direction for each component Magnitude equals “1” “i” represents the x direction, “j” represents y

direction (“k” will represent the z direction)

X

Y

r

q

Dx

Dy jyixr ˆ)(ˆ)(

Unit Vectors

Once the components for two or more vectors are determined: To add the vectors, simply add like components of

each vector

jcicC

jbibB

jaiaA

yx

yx

yx

A

B

C

For motion when direction is changing…we can use 1-D motion to analyze the components

…then add the components to get the resulting motion

R

jcbaicbaCBAR

then

yyyxxx )()(

Determining Magnitude

To find the magnitude, r, of the Resultant vector, we use the Pythagorean Theorem:

X

Y

Dx

Dy 22

222

,

yxr

or

yxr

r

Determining Direction

We can use the tangent function to determine the value of : q

X

Y

Dx

Dyr

q x

y

orx

y

1tan

,

tan

If using a calculator, be sure to set degrees or radians as appropriate

Relative Motion

Try this…

How fast must a truck travel to stay directly beneath and airplane that is moving 105 km/hr at an angle of 25 degrees to the ground?

X

Y

V=105 kph

=25q o

Vtruck=?

hr/kmcos

cosvvtruck9525105

Try this…

A ranger leaves his base camp for a ranger tower. He drives on a heading of 125o for 25.5 km and then drives at a heading of 65o for 41.0 km. What is the displacement from the base camp to the tower?

Summary

Analyzing Vectors…1. Resolve vectors into components “i”, “j”, “k”2. Analyze Components (addition, motion…)3. Add components to get new Resultant

We will use these steps to help us analyze motion in 2-D…

Projectile Motion

The most common example of an object which is moving in two-dimensions is a projectile

A projectile is an object upon which the only force acting is gravity

Projectile Motion- motion observed by any object which once projected is influenced only by the downward force of gravity.

(provided that the influence of air resistance is negligible)

Projectile Motion There are a variety of examples of projectiles:

an object dropped from rest is a projectile an object which is thrown vertically upwards is also a projectile an object is which thrown upwards at an angle is also a projectile

Projectile Motion

Here is a typical projectile… a baseball thrown in the air

Remember… Any vector can be resolved into two perpendicular component vectors

Velocity

VerticalComponent

HorizontalComponent

x

y

Projectile Motion

Once the ball leaves your hand … it is only influenced by the acceleration of gravity

This is the definition of projectile motion

Velocity

VerticalComponent

HorizontalComponent

x

y

g

Projectile Motion

Since gravity acts along the vertical axis, the horizontal component of velocity is unaffected by gravity

Horizontal Projectile

Non-Horizontal Projectile

Resolving Components

If we know the magnitude and direction of the projectile’s velocity, you can find the x and y components:

Velocity

VerticalComponent

HorizontalComponent

x

y

g

v

q

sinvv

cosvv

y

x

Projectile Motion

In the horizontal direction, acceleration is zero and velocity is constant

Velocity

VerticalComponent

HorizontalComponent

x

y

g

txvx

:direction- xin the

nt displacemefor solve To

Projectile Motion

In the vertical direction, acceleration is constant (g)

All the equations derived for 1-D motion apply.

Velocity

VerticalComponent

HorizontalComponent

x

y

ggtvvHow

gttvyyHow

yiyf

yiif

fast... 2

1 far... 2

Projectile Motion

Vertical and Horizontal velocities are independent

Time is the factor that relates the two components

To solve projectile motion problems.. You must use the given info to find the time, t.

Velocity

VerticalComponent

HorizontalComponent

x

y

g

Try this…

You throw a ball horizontally at 20 m/s from a 10 m tower. How far will the ball go before it hits the ground?

Projectile Path

Since the vertical distance component is related to time squared (y=1/2gt2), projectiles will follow a parabolic path. (note t = x/vcos)

Demo…

A projectile is shot with velocity, v, at an angle, q. Assuming no air resistance…

1. How far, x, will it travel from the launch site?2. What is the maximum distance it will travel?3. What angle will give it the maximum distance?4. At what two angles will give it travel the same

distance?5. What is the maximum height at each angle?

Do…

A cannon will shoot a projectile with a muzzle velocity of 320 m/s.

1. What is the cannon’s maximum range? (10450m) 2. What angles will give a range of 3000. m? (8.3o, 81.7o)

3. What is the max height reached at each angle? (109m, 5100m)

Projectile Path

Without Gravity… a projectile would follow the dotted path

With Gravity… the projectile falls beneath this line the same vertical distance that it would if dropped from rest

Without Gravity… a projectile would follow the dotted path

With Gravity… the projectile falls beneath this line the same vertical distance that it would if dropped from rest

What keeps a satellite in orbit?

SatellitesV < 8000 m/s V almost 8000 m/s

V = 8000 m/s V > 8000 m/s

8000m/s at an altitude of about 190 km

A satellite is simply a projectile that is constantly falling toward earth

Summary

Analyzing 2-D Motion…1. Resolve displacement, velocity, or acceleration

vectors into components 2. Use 1-D equations of motion to determine motion

along each component3. Add components to get Resultant motion

Projectile motion Motions in x and y directions are independent Time relates the motion along each axis and must

be determined

What we know…

Add, subtract, and resolve displacement and velocity vectors, so we can: determine components of a vector along two perpendicular axes determine the displacement and location of a particle relative to

another determine the change in velocity of a particle or the velocity of

one particle relative to another Understand the motion of projectiles in a uniform

gravitational field, so we can: Determine the horizontal and vertical components of velocity and

position as functions of time Analyze the motion of a projectile that is projected with an

arbitrary initial velocity

Questions?