“Seeking…“Weight proportional to height.”

What’s that supposed to mean?

Here’s a fairly common trait sought after

in the personals:

Is that a desirable trait?

Interlude:

ProportionalitiesA

B CD

wx

A z

?

1.A/w2.A/x3.B/x4.C/x5.D/x6.D/w

y

z

A and z are bases of their respective (isosceles) triangles, i.e, they are “corresponding sides.” D is an“altitude” and so corresponds to x.

R

A a

?

1.R/r2

2. R2/r2

3. R2/r4. R/r5. r/R

rThe ratio of surface areas

A

a

The ratio of the sides of the square areas S/s = R/r since S is to R as s is to r.However the areas A = S2 and a = s2.S

s

height, h

weight, W

Consider this block weighing “W”

height, 2hThis stack of 2 blocks

weighs how much?

2W

Are these blocksin proportion?

To scale proportionally

height, h

weight, W

And this double-sized block weighs

1. 2W 3. 6W 5. 10W2. 4W 4. 8W 6. 12W

More generally,

h

wL 2h

2w2L

originalvolume =hwL

newvolume

=(2h)(2w)(2L)

= ( 8 )hwL

= ( 8 )( )original volume

Is weight meant to be proportional to height?

Weight (Height)3

Each 1% increase in height shouldcorrespond to a (1.01)3 = 1.03

3% increase in weight

5% increase in height (5’4” 5’7”) 15.2% gain in weight

10% increase in height (5’10” 6’5”) 30% gain in weight

Easy topush

Hard topush

To produce the same amount of acceleration, need to push the car much harder!

How much?

Twice as much mass requires twice the force!

100 as much mass requires 100 the force!

F m

If a lot of others join in, each contributingan additional force, the net force will

be greater, and the acceleration greater.

F = ma

F a F mtogether with

can be combined consistently to give

Newton’s 2nd Law

The Toyota FT-HS is a 400-horsepower hybrid sports car

 its 3.5-liter V6 engine provide an acceleration of

0-to-60-miles-an-hour in 4 seconds

Unveiled at yesterday’s International Auto Show in Detroit:

acceleration, a = v t

+60 miles/hour4 seconds

= 15 miles/hoursec

+60 miles/hour4 seconds

= 15 miles/hoursec

Notice 60 miles/hour is the same as

60 = 316800 mileshour

5280 feet1 mile

feethour

316800 =feethour

1 hour3600 seconds

feetsec

88

+60 miles/hour4 seconds

= 22 feet/secsec

So alternately

or 22 ft/sec2

Softball6.5-7.0 oz.

Tennis ball2 ounces

(57 grams)

14 lb. medicine ball

32 timesheavier than a

softball!

112 timesheavier than a tennisball!

If all 3 balls were released simultaneously, from the same height, they would reach ground 1. tennis ball first, medicine ball last. 2. medicine ball first, tennis ball last. 3. together at about the same time.

A large boat whose maximum speedin still water is v1, tows a smaller boat

whose maximum speed is the smaller v2.across the lake. If both outboard motorsrun together at full bore, the speed thatthey travel together with will be

1. the smaller value v1.2. the larger value v2.

3. somewhere between v1 and

v2.

agravity gFalling objects accelerate down at

32 ft/sec2

9.8 m/sec2

more precisely:at sea level 9.80621 m/sec2

32.1725 ft/sec2

16.0 km (10 miles) above earth’s surface agravity drops to about 9.75 m/sec2

32.5 km (20 miles) above earth’s surface agravity drops to about 9.70 m/sec2

even skydivers experience a 9.8 m/sec2

even commercial jet carriers experience agravity only 1% under the value at sea level!

North Pole agravity = 9.832San Francisco agravity = 9.800Denver agravity = 9.796

The gravitational force on an object decreases by about a millionth for every

3 meter (~10 feet) gain in elevation.

An individual with a 50 kilogram mass weighs 500 Newtons (110 pounds) in

New York City; but ~0.25 newton  (1 ounce) less in mile-high Denver.

If you drop an object (assuming itsair resistance is negligible) it accelerates downward at g=9.8 m/sec2. If insteadyou throw it downward, its downwardacceleration after release is

A. <gB. =gC. >g

Inclined Plane

12

22

32

42

52

A ball rolling down an inclined plane has constant acceleration

Is the acceleration of the ball down the ramp 9.8 m/s2?

A) yes B) no

No! This ball is not in free fall. Gravity alone does not act on it.

The inclined plane provides a forceof support which affects the motion!

A ball is thrown straight up and falls back to the ground. Which of the following is true about its velocity v and its acceleration a at the highest point in its path?

A) v = 0 and a = 0

B) v 0, but a = 0

C) v = 0, but a 0

D) v 0 and a 0

E) cannot be determined

At the highest point in its path, the ball momentarily comes to a stop, and so its velocity is zero.

However, since the ball is in free fall, its acceleration is g = 9.8 m/s2 (at every moment).

A ball is dropped from rest, and a bullet shot out of a gun, straight down. Neglecting air resistance, which has the greater acceleration just before hitting the ground?

A) the ball

B) the bullet

C) both have the same acceleration

The acceleration of gravity does not depend on the mass or the speed of the object in free fall!

The maximum velocity, v, an object reaches falling freely from rest, is

directly proportional to the time, t, of its fall: v t .

1) TRUE 2) FALSE

How fast is an object moving at the endof a one second fall?

How far does an object in freefall dropin one second?