Question

A 27 kg rock starting from rest free-falls through a distance of 18.0 m with no air resistance. Find the momentum change of the rock caused by its fall and the resulting change in the magnitude of the earth’s velocity. The earth’s mass is 6.0 × 10^24 kg. Show all work, assuming the rock–earth system is closed!

***Not sure how to begin! Thank you so much! :)

Answer 1

Okay. Cool.

So, I got a rock. That is gonna hit the earth the earth with some velocity.
And because of the velocity, it's gonna have some momentum, right?

Answer 2

yes:)

Answer 3

Now, Momentum = P = M x V

You know the mass. You need the velocity. You know the rock thing fell through 18 metres, free-fall.
Can you find the rock's velocity right before it hit the ground?
(Hint: Newton's equations ;) )

Answer 4

okay, erm soo plugging in, i get
18=(6.0*10^24) * V ? :/

and not sure which one to use for newtons equations?

Answer 5

\(s = \frac{v^2-u^2}{2a}\)
S = Distance
v = final velocity, we need to find that
u = Initial speed. Can this be zero? ;)
a = Acceleration. Gravity is the accelerating force. So A = what?

Answer 6

What do you think about this one instead? :]

Answer 7

okay:)

would it be like this?

18=v^2-0^2/2(27)

18= v^2/ 2(27)?

not sure if 27 is the acerbating force? :/

Answer 8

What is 27?

Answer 9

ohh that's the starting weight? darn.. not sure how to find the accelerating force :( how would i find that?

Answer 10

I see that it's the MASS. It's not really a force.

Also, you would remember, in a free-falling body, where air-resistance is not being considered, you don't really need the mass. right?

Remember, acceleration due to gravity = g = 9.8 m/sec^2 ??
Think we can use it here? ;)

Answer 11

ohh okay! got it:)

so 18=v^2/2(9.8)

18=v^2/19.6
18*19.6=v^2
v^2=352.8
v=18.7829... ?

Answer 12

Ohkay so far! Seems good.

So, what's the momentum then?

Answer 13

yay! so this is where i use the p=m*v right?

so momentum=p=27*18.7829=507.14? :/

Answer 14

Yep yep.

Ok.. Now. Imagine this. The rock hits the ground. Comes to rest. But the entire momentum it was carrying is transferred to the earth.

This momentum P is added to earth momentum. And the change in earth's velocity is this momentum P divided by earth's mass.

Can we find the change in earth's velocity, then?

Answer 15

okay, ermm like 507.14-(6.0*10^24)? not sure:(

Answer 16

ohh wait 507.14/9.8 ?

Answer 17

and the change would be 51.7489.... ?

Answer 18

Lol No.

Just DIVIDE the momentum by earth's mass.

\(\Delta P / M= \Delta V \)

Answer 19

The number would be very small, but guess what - that's what happens.
Even if you drop a very heavy object on the ground, you don't really feel our earth's speed changing. Right? ;)

Answer 20

ohhh okay, so 507.14/6.0*10^24?

Answer 21

and it becomes 5.0714E26?

Answer 22

"5.0714E26"?

Answer 23

im not sure, that's what my calculator says when i enter those numbers? :/ what should it be? :/

Answer 24

\(\frac {507.14} {6.0 \times 10^{24}} =\frac {507.14 \times 10^{-24}} {6.0} \)

So you can simply divide 507.14 by 6.0 and leave the answer with 10^(-24).

Answer 25

ohh okay! and so the change in earth's velocity would be 10^-24? :O

Answer 26

and so the problem is complete? :O

Answer 27

Nah. NOOO!!!!

You need to multiply that with the 507.14/6.0

Answer 28

ohhh okay!! haha so 84.523/10^-24=8.45? :/ i don't know, but my calculator says E25 at the end? :/

Answer 29

The heck witht he calculator! :P Let's do it this way:

\(\large \frac {507.14} {6.0 \times 10^{24}} =\frac {507.14 \times 10^{-24}} {6.0}= \frac {507.14 } {6.0}\times 10^{-24} = 84.523 \times 10^{-24}\)

Answer 30

Understandable? Don't forget to mention the right units.

Answer 31

hahaha okay:P

and yes:) so that would just be the change? and the units would be kg/s?

Answer 32

and so now the problem is done? :O

Answer 33

Yep!

Answer 34

ooh yay!! thank you so much!! :D

Answer 35

M/sec!!!

Speed is NOT kg per sec. It's distance per unit time!!!!

Answer 36

ohh okay!! thank you:D

Answer 37

Okay. :)

Answer 38

oh so it asks for resulting change in magnitude, which value was that again? :O

Answer 39

The one that you found it.
The change in magnitude of the velocity is caused by the change in the mome
ntum, which in turn, was caused by the falling rock which came to rest and transferred all its momentum to the earth.

Answer 40

okay yay! :) thank you!!