10. Free fall

Free fall acceleration: g=9.8m/s2

Using general equations:

atvv

attvxx

0

2

00 2

Substitute:

yx

ga

To derive the following equations:

gtvv

gttvyy

0

2

00 2

2

2

0

20

20

vv

t

yv

vvyyg

2

2

0

20

20

vv

t

xv

vvxxa

1

Example: A rocket is fired at a speed of 100 m/s straight up. (Neglect air resistance.)a)How long does it take to go up to the highest point?b)What is the maximum height?c)How long does it take to return back (go up and fall back down)?d)What is the speed of the rocket just before it falls down?

gvt

tgttv

gttvyy

c 0

2

0

2

00 2

0

20

2

0yy

0000 2 vvgvgvgtvv c

Solution:

0v

./1000 smv

a) At the highest point

;0 000 gvtgtvgtvv a ssmsmta 2.108.9/100 2

c) When it comes back

;

222

20

20

0

2

00max g

vgvgtv

gttvyyh a

aa

Given:

m

sm

smh 510

8.92

/1002

2

max

b) The maximum height is

d) The speed it falls down with is

stt ac 4.202

2

The trajectory of an object projected with an initial

velocity at the angle above the horizontal

with negligible air resistance.0v

0

11. Projectile Motion

sin

cos

00

00

vv

vv

y

x

smga

constvv xx

/8.90

3

Shoot the monkey (tranquilizer gun)

A zookeeper shoots a tranquilizer dart to a monkey that hangs from a tree. He aims at the monkey and shoots a dart with an initial speed v0.

The monkey, startled by the gun, lets go immediately. Will the dart hit the monkey?

A. Only if v0 is large enough.

B. Yes, regardless of the magnitude of vo.

C. No, it misses the monkey.

If there is no gravity, the dart hits the monkey…

If there is gravity, the dart also hits the monkey!

4

If there is no gravity, the dart hits the monkey…

Continued

5

If there is gravity, the dart also hits the monkey!

Note, that it takes the same amount of time to hit the monkey as in the no gravity case!

Continued

6

This might be easier to think about…

2

0

2

1gty

tvx

For the bullet:

2

0

2

1gty

xx

For the monkey:

Continued

7

Example: A rocket is fired at a speed of 100 m/s at an angle 30° above the horizontal . (Neglect air resistance.)a)What are the initial values of the x and y components of the speed?b)How long is the rocket be in the air?c)What is the distance between the lunch and landing points (assuming that these points have the same altitude)?

Solution:

smv /1000 Given:

a) smsmvv

smsmvv

y

x

/0.5030sin/100sin

/6.8630cos/100cos

00

00

b) From the example on slide 2, question (c) follows that

;2 0 gvt y ssmsmt 2.108.9/502 2

c) The horizontal motion is uniform, and the distance is

mssmtvd x 8832.10/6.860

0v

xv0

yv0

x

y

8

30

12. Circular motion v

r

d

x

y

r

r

varad

2

frequencyangular - 2

frequency - 1

πf

Tf

Definitions:

For any circular motion there is radial (centripetal) acceleration:It is directed along radius and to the center.

For uniform circular motion (v=const):

rfrT

r

t

dv

22

T – period (time it takes to return to the same point)

rarad2

9

t

va

t

0

tan lim

22tan radaaa

Optional information (if you are curious):

If circular motion is not uniform (v ≠ const) then there is tangential acceleration:

The tangential acceleration is perpendicular to the radial acceleration, and the total acceleration is equal to:

12a. Circular motion (continues)

10

Example: Period of a satellite motion

g

R

kmR

smg

6400

/8.9 2

ga

R

va

2

gR

v

2

gRv g

R

v

RT

22

min83

min60

500050008002

/8.9

/100064002

2

sss

sm

kmmkmT

skmsmkmmkmsmgRv /8/108/10006400/8.9 32

11

12

2,1

2,1

2

1

?

?

20

40.0

20.0

rada

v

T

rpmf

mr

mr

sf

TTss

f 31

3

1

60

min1

min

120 21

sms

mv

sms

mv

/84.03

40.02

/42.03

20.02

2

1

T

r

t

dv

2

Example: Two balls attached to a string as shown at 0.20 m and 0.40 m from the center move in circles at a uniform frequency of 20 rpm. What are their periods, linear speeds, and radial accelerations?

rfarad22

22

2

22

1

/76.140.03

2

/88.020.03

2

smms

a

smms

a

rad

rad

12