LCROSS crashes into the Moon

Image credit:NASA

LCROSS studied plumes created by impact trying to measure composition of the ejecta and estimate amount of water ice. Then, in 4 min, it also crashed.

Centaur rocket stage is released from the circular orbit 86400 km above the lunar surface. Find the speed with which the stage crashes into the Moon. Moon’s mass is 7.3x1022 kg, radius 1740 km. Gravitational constant is 6.67x10-11 N m2 kg-2.

How much energy is released in the crash if the rocket stage has mass of 2300 kg? Find TNT equivalent (1 kg TNT = 4x106 J)

2.365 km/s; 6.3x10^9 J

Compare to most powerful explosions

Tsar Bomb 196150 MT

Castle Bravo 195415-25 MT

.

<http://eol.jsc.nasa.gov/debrief/Iss020/topFiles/ISS020-E-9048.htm> ( 10/15/2009 22:06:18)

All forces are

CONSERVATIVE or

NON-CONSERVATIVE

Chapter 8

A force is conservative if:

The work done by the force in going from to is independent of the path the particle follows

The work done by the force when the particle goes from around a closed path, back to , is zero.

or

1r

2r

1r

1r

Non-conservative: doesn’t satisfy the above conditions

Theorem: if a force can be written as the gradient (slope) of some scalar function, that force is conservative.

U(x) is called the potential energy function for the force F

If such a function exists, then the force is conservative

1D case:dx

dUFx

conW does NOT depend on path!

)]()([ 12

2

1

2

1

xUxU

dxdx

dUdxFW

x

x

x

x

xcon

If Fx(x) is known, you can find the potential energy function as

CdxxFxU x )()(

dx

dUFx

Work-energy theorem:

nccontotal WWWKK 21212112

ncWUUKK 211212

ncWUKUK 211122

const,0If 112221 UKUKW nc

Energy conservation law!

Then use

0if, 211122 ncWUKUK

ncWUKUK 211122 or

A strategy: write down the total energy E = K + U at the initial and final positions of a particle;

Examples

Force of gravity

Spring force

y

CmgyyU )(

xx0

Cxxk

xU

2

)()(

20

)( 0xxkFx

mgFy

A gun shoots a bullet at angle θ with the x axis with a velocity of magnitude Vm. What is magnitude of the velocity when the bullet returns to the ground? Hits the target at height H above the ground? How high it will go?

Note: motion is 2D, but U(y) = mgy is still a function of only one coordinate y.

Potential Energy Diagrams

• For Conservative forces can draw energy diagrams

• Equilibrium points– If placed in the

equilibrium point with no velocity, will just stay (no force) 0 dx

dUxF

Fx >0

a) Spring initially compressed (or stretched) by A and released;b) A block is placed at equilibrium and given initial velocity V0

Stable vs. Unstable Equilibrium Points

The force is zero at both maxima and minima but…– If I put a ball with no velocity there would it stay?– What if it had a little bit of velocity?

A block of mass m is (not) attached to a vertical spring, spring constant k.

A

If the spring is compressed an amount A and the block released from rest, how high will it go from its initial position?

A particle is moving in one direction x and its potential energy is given by U(x) = ax2 – bx4 . Determine the force acting on a particle.Find the equilibrium points where a particle can be at rest. Determine whether these points correspond to a stable or unstable equilibrium.

Block of mass m has a massless spring connected to the bottom. You release it from a given height H and want to know how close the block will get to the floor. The spring has spring constant k and natural length L.

H

y=0

L

Water SlideWho hits the bottom with a faster speed?

http://curvebank.calstatela.edu/brach/brach.htm

The curve of fastest descent

Cycloid

Inverted cycloid:Brachistochrone

H

Roller CoasterYou are in a roller coaster car of mass M that

starts at the top, height H, with an initial speed V0=0. Assume no friction.

a) What is the speed at the bottom?b) How high will it go again?

c) Would it go as high if there were friction?

H

Roller Coaster with FrictionA roller coaster of mass m starts at rest at height y1 and falls down the path with friction, then back up until it hits height y2 (y1 > y2).

Assuming we don’t know anything about the friction or the path, how much work is done by friction on this path?

z

zyxUF

y

zyxUF

x

zyxUF zyx

),,(;),,(

;),,(

Several dimensions: U(x,y,z)

Compact notation using vector del, or nabla:

kz

jy

ix

UF

,

Another notation:rd

dUF

Partial derivative is taken assuming all other arguments fixed

Geometric meaning of the gradient :UDirection of the steepest ascent;

Magnitude : the slope in that directionU

:UF

Direction of the steepest descent

Magnitude : the slope in that direction F

http://reynolds.asu.edu/topo_gallery/topo_gallery.htm

)]()([ 12 rUrUW con

Ifrd

dUF

)()( 12

)(

)(

2

1

rUrUdUrdrd

dUrdFW

rU

rUL

z

zyxUF

y

zyxUF

x

zyxUF zyx

),,(;),,(

;),,(

or

then

Work-energy theorem:

nccontotal WWWKK 21212112

ncWUUKK 211212

ncWUKUK 211122

const,0If 112221 UKUKW nc

Energy conservation law!

Checking if the force is conservative

bFaF yx ;

byFaxF yx ; )();( ygFxfF yx

bxFayF yx ; bFaxyF yx ;

Checking if U(x,y) is a potential energy for any force

xy

yxU

yx

yxU

),(),( 22

Find the velocity of the block at points 0,B, and DRepeat if there I friction on the table up to x = B

PowerPower is a rate at which a force does work

If work does not depend on time(or for average power):

t

WP

Otherwise: vFdt

rdF

dt

WP

Even if instantaneous power depends on time, one can talk about the average power

How many joules of energy does 100 watt light bulb use per hour? How fast would a 70-kg person have to run to have that amount of energy?

Power could also define the rate at which any form of energy is spent, not only mechanical