Physics Walker Chapter 4 Notes

Copyright 2010 Pearson Education, Inc.

Chapter 4

Two-Dimensional

Kinematics


Units of Chapter 4

Motion in Two Dimensions

Projectile Motion: Basic Equations

Zero Launch Angle

General Launch Angle

Projectile Motion: Key Characteristics


4-1 Motion in Two Dimensions

motion along a

straight line with

constant velocity

=

= cos = sin

= = 0


Vector equations

=

is equivalent to

= =

Two vectors are equal when their components

are equal


4-1 Motion in Two Dimensions

Rule: motion in the x- and y-directions is solved

separately !

Vector notation:

= + +

; = +

Component notation: = , , = ,


4-2 Projectile Motion: Basic Equations

Assumptions:

ignore air resistance

Acceleration due to gravity g = 9.81 m/s2,

downward

ignore Earths rotation

If y-axis points upward, acceleration in x-direction is

zero and acceleration in y-direction is -9.81 m/s2

= = = .



The acceleration is independent of the direction

of the velocity:



These, are the basic equations of projectile motion:

X direction:

= + = =

Y direction:

= +

12

2

=

= ( )

X Motion with

constant velocity

Y Motion with

constant acceleration

A problem is set by initial conditions: , , ,


4-3 Zero Launch Angle

Launch angle: direction of initial velocity with

respect to horizontal, =



In this case, the initial velocity in the y-direction

is zero. Here are the equations of motion, with

= , = , = , = :

X direction:

= = =

Y direction:

= 12 2

=

= ( )

Motion with constant Drop from height

Question 4.2 Dropping a Package

You drop a package from

a plane flying at constant

speed in a straight line.

Without air resistance, the

package will:

a) quickly lag behind the plane

while falling

b) remain vertically under the

plane while falling

c) move ahead of the plane while

falling

d) not fall at all

You drop a package from

a plane flying at constant

speed in a straight line.

Without air resistance, the

package will:

a) quickly lag behind the plane

while falling

b) remain vertically under the

plane while falling

c) move ahead of the plane while

falling

d) not fall at all

Both the plane and the package have

the same horizontal velocity at the

moment of release. They will maintain

this velocity in the x-direction, so they

stay aligned.

Follow-up: what would happen if air resistance is present?

Question 4.2 Dropping a Package

Question 4.3a Dropping the Ball I

From the same height (and

at the same time), one ball

is dropped and another ball

is fired horizontally. Which

one will hit the ground

first?

a) the dropped ball

b) the fired ball

c) they both hit at the same time

d) it depends on how hard the ball

was fired

e) it depends on the initial height

From the same height (and

at the same time), one ball

is dropped and another ball

is fired horizontally. Which

one will hit the ground

first?

a) the dropped ball

b) the fired ball

c) they both hit at the same time

d) it depends on how hard the ball

was fired

e) it depends on the initial height

Both of the balls are falling vertically under the influence of

gravity. They both fall from the same height. Therefore, they will

hit the ground at the same time. The fact that one is moving

horizontally is irrelevantremember that the x and y motions are

completely independent !!

Follow-up: is that also true if there is air resistance?

Question 4.3a Dropping the Ball I



This is the trajectory of a projectile

launched horizontally: = ( )

Landing = at



Eliminating t and solving for y as a function

of x:

This has the form y = a + bx2, which is the

equation of a parabola.

The landing point can be found by setting

y = 0 and solving for x:


4-4 General Launch Angle

Initial conditions:

= , = , = cos , = sin


4-4 General Launch Angle

In general, v0x = v0 cos and v0y = v0 sin

This gives the equations of motion:

X direction:

= ( cos ) = cos

Y direction:

= ( sin )

12

2

= sin

= (sin )


4-5 Projectile Motion: Key Characteristics

Hang time (time spent in the air):

= sin

= sin

=



Height (max):

=( sin )

Which of the

three punts

has the

longest hang

time?

Question 4.4a Punts I

d) all have the same hang time

a b c

h

Which of the

three punts

has the

longest hang

time? d) all have the same hang time

a b c

h

The time in the air is determined by the vertical motion!

Because all of the punts reach the same height, they all

stay in the air for the same time.

Follow-up: Which one had the greater initial velocity?

Question 4.4a Punts I



Range: the horizontal distance a projectile

travels

If the initial and final elevation are the same:

a b d c

Question 4.5 Cannon on the Moon

For a cannon on Earth, the cannonball would follow path 2.

Instead, if the same cannon were on the Moon, where g =

1.6 m/s2, which path would the cannonball take in the same

situation?

a b d c

The ball will spend more

time in flight because

gMoon < gEarth. With more

time, it can travel farther

in the horizontal

direction.

For a cannon on Earth, the cannonball would follow path 2.

Instead, if the same cannon were on the Moon, where g =

1.6 m/s2, which path would the cannonball take in the same

situation?

Follow-up: which path would it take in outer space?

Question 4.5 Cannon on the Moon


Summary of Chapter 4

Components of motion in the x- and y-directions can be treated independently

In projectile motion, the acceleration is g

If the launch angle is zero, the initial velocity has only an x-component

The path followed by a projectile is a parabola

The range is the horizontal distance the projectile travels

Physics Walker Chapter 4 Notes

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