# Chapter 4

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14 Νοε 2013 (πριν από 4 χρόνια και 6 μήνες)

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In this chapter we will learn about
kinematics
(displacement, velocity,
acceleration) of a particle in two dimensions (plane).

All four kinematic equations are still true, only in vector form

Superposition principle

treat vector components independently

Projectile motion

Chapter 4:

Chapter 4.1, 4.2, 4.3

4.4 and 4.5 will be combined with chapter 6, skip 4.6

Homework: 4 (due Thursday, Sept. 20):

QQ3, AE1, AE4, 3, 5, 7, 9, 16, 21, 62

Remember:

Homework 3.1 is due today (Sept. 13) and Homework 3.2 is due Tuesday (Sept. 18)

You throw two rocks straight up in the air. One reaches a height of
10m, the other a height of 20 m. Which one will be in the air
longer? (Ignore air resistance).

A. 10 m.

B. 20m.

C. The same time.

E. Depends on weight.

Displacement in a
plane

The displacement vector
r
:

i
f
r
r
r

Displacement is the straight line between the final
and initial position of the particle.

That is the vector difference between the final and
initial position.

Average Velocity

Average velocity
v
:

t
r
v

Average velocity
: Displacement of a particle,

r
, divided by time
interval

t.

Instantaneous
Velocity

Instantaneous velocity
v
:

dt
r
d
t
r
v
t

0
lim
Instantaneous velocity
v
: Limit of the average velocity as

t
approaches zero.

The instantaneous velocity equals the derivative of the position
vector with respect to time.

The magnitude of the instantaneous velocity vector

is called the
speed

(scalar)

v
v

Average
Acceleration

Average acceleration:

t
v
t
t
v
v
a
i
f
i
f

Average acceleration
: Change in the velocity

v
divided by the
time

t during which the change occurred.

Change can occur in direction and magnitude!

Acceleration points along
change in velocity

v
!

Quick quiz: Is it possible that a particle with constant
speed

experiences an acceleration
?

A. Yes.

B. No

Instantaneous
Acceleration

Instantaneous acceleration:

Instantaneous acceleration
: limiting value of the ratio

as

t goes to zero.

Instantaneous acceleration equals the derivative of the velocity
vector with respect to time.

dt
v
d
t
v
a
t

0
lim
t
v

Two
-
dimensional motion with
constant

acceleration
a

Trick 1:

The equations of motion (kinematic equations) we derived
before are still valid, but are now in vector form.

Trick 2 (Superposition principle):

Vector equations can be broken down into their x
-

and y
-

components. Then calculated independently.

y
x
r
j
y
i
x
r

Position vector:

Velocity vector:

y
x
y
x
v
v
v
j
v
i
v
v

Two
-
dimensional motion with
constant

acceleration

t
a
v
v
i
f

Velocity as function of time:

2
2
1
t
a
t
v
r
r
i
i
f

Position as function of time:

t
a
v
v
t
a
v
v
y
yi
yf
x
xi
xf

2
2
2
1
2
1
t
a
t
v
y
y
t
a
t
v
x
x
y
yi
i
f
x
xi
i
f

Similarly, the other two kinematic equations are now in vector form (and can
also be broken up into their x
-

and y
-
components:

)
(
2
2
2
i
f
i
f
r
r
a
v
v

t
v
v
r
r
f
i
i
f
)
(
2
1

A melon truck brakes
right before a ravine and
looses a few melons. The
melons skit over the edge
with an initial velocity of
v
x

= 10.0 m/s.

(a)
Determine the x
-

and y
-
coordinates of the particle at any time t
and the position vector
r

at any time t.

(b)
Determine the x
-

and y
-
components of the velocity at any time
and the total velocity at any time.

(c)
Calculate the impact angle, the velocity and the speed of the
melons as they hit the bottom of the ravine.

Black board example 4.1

The melons hit the bottom of the ravine after 5 s.

Projectile motion

Two assumptions:

1.
Free
-
fall acceleration
g

is constant.

2.
Air resistance is
negligible.

-

The path of a projectile is a
parabola

(derivation: blackboard and see book).

-

Projectile leaves origin with an initial velocity of
v
i
.

-

Projectile is launched at an angle
q
i

-

Velocity vector changes in magnitude and direction.

-

Acceleration in y
-
direction is
g
.

-

Acceleration in x
-
direction is 0.

yf yi
v v gt
 
2
1
2
f i yi
y y v t gt
  
xi
xf
v
v

t
v
x
x
xi
i
f

Acceleration in x
-
direction is 0.

Acceleration in y
-
direction is
g
.

(Constant velocity)

(Constant acceleration)

Projectile motion

Superposition

of motion
in x
-
direction and
motion in y
-
direction

A battleship simultaneously fires two shells at enemy ships.

If the shells follow the parabolic trajectories shown, which ship gets hit first?

A.

B.

C. Both hit a the same time.

Quick quiz:

Hitting the bull’s eye. How’s that?

Demo.

i
-
clicker:

You shoot a bullet with 600 m/s straight at a target that is 600 m
away. By how much does the bullet ‘drop’ during is flight?

A.
0 m

B.
0.49 m

C.
0.98 m

D.
4.9 m

E.
9.8 m

600 m

A rescue plane drops a
package to stranded
explorers. The plane is
traveling horizontally at
40.0 m/s and is 100 m
above the ground.

(a)
Where does the package strike the ground relative to the point at which it was
released.

(b)
What are the horizontal and vertical components of the velocity of the
package just before it hits the ground? What is the speed of the package as it
hits the ground?

(c)
Where is the plane when the package hits the ground? (Assume that the plane
does not change its speed or course.)

Black board example 4.2

Quick review

Kinematics (displacement, velocity, acceleration) of a particle in two dimensions.

All four kinematic equations are still true, only in vector form:

Superposition principle

treat vector components independently!!

solve components independently, time connects equations for x
-

and y
-

components

Projectile motion:

t
a
v
v
i
f

2
2
1
t
a
t
v
r
r
i
i
f

t
a
v
v
t
a
v
v
y
yi
yf
x
xi
xf

2
2
2
1
2
1
t
a
t
v
y
y
t
a
t
v
x
x
y
yi
i
f
x
xi
i
f

)
(
2
2
2
i
f
i
f
r
r
a
v
v

t
v
v
r
r
f
i
i
f
)
(
2
1

1
2
1
2
( )
( )
f i xi xf
f i yi yf
x x v v t
y y v v t
  
  
2 2
2 2
2 ( )
2 ( )
xf xi f i
yf yi f i
v v a x x
v v a y y
  
  