Inelastic collision on a frictional surface

[aesnix]

[SOLVED] Inelastic collision on a frictional surface

 

[aesnix]

Sorry I clcked the wrong button and sent it out before typing anything... I didn't even catch up on the 30-min editting time. Please forgive my slow typing...

Homework Statement

A wood block with mass M was laid on a frictional surface with coefficient of kinetic friction u. A bullet with mass m flew towards the wook block with constant speed v; it shot into the block and moved along with the block for a distance before coming to rest. Given the force that the bullet experienced in the block was f, try to find out the distance D the block moved on the surface, and the distance d the bullet moved in the block.

Homework Equations

Work = Force x Distance (constant parallel force)
inelastic collision: m1 x v0 = (m1 + m2) x v

The Attempt at a Solution

I assumed that the collision was fast enough that the block didn't move apprecialbly during the collision, thus:
mv = (M+m)v'....so v'=mv/(M+m)

and, with energetic considerations:
for bullet m: 1/2mv^2 - fd = 1/2mv'^2
for block M: 1/2Mv'^2 - u(M+m)gD = 0

I know my assumption may well be wrong, but I had no clue of doing it in another way. I didn't include any heat that was lost either, but I don't know how. I guess I don't understand the characteristics of frictional force very well. The wiki says the heat lost during a movement on a frictional surface is f x d... uh, why?

According to Newton's 3rd law, I thought that if the surface doesn't move, then the work that was "supposed to done by an object on the surface" is lost as heat... but what if "the surface," as in the above case, IS moving? Also, if the Earth served as "the surface", wouldn't it acquire a very very small acceleration when an object slides on it?

Why, then. does the energy become heat but not the Earth's kinetic energy?

 

[alphysicist]

Hi aesnix,

Sorry I clcked the wrong button and sent it out before typing anything... I didn't even catch up on the 30-min editting time. Please forgive my slow typing...

Homework Statement

A wood block with mass M was laid on a frictional surface with coefficient of kinetic friction u. A bullet with mass m flew towards the wook block with constant speed v; it shot into the block and moved along with the block for a distance before coming to rest. Given the force that the bullet experienced in the block was f, try to find out the distance D the block moved on the surface, and the distance d the bullet moved in the block.

Homework Equations

Work = Force x Distance (constant parallel force)
inelastic collision: m1 x v0 = (m1 + m2) x v

The Attempt at a Solution

I assumed that the collision was fast enough that the block didn't move apprecialbly during the collision, thus:
mv = (M+m)v'....so v'=mv/(M+m)

and, with energetic considerations:
for bullet m: 1/2mv^2 - fd = 1/2mv'^2
for block M: 1/2Mv'^2 - u(M+m)gD = 0

Since the bullet and block are moving together, I think this needs to be:

$$\frac{1}{2}(M+m) v'^2 - u (M+m) g D =0$$

I know my assumption may well be wrong, but I had no clue of doing it in another way. I didn't include any heat that was lost either, but I don't know how. I guess I don't understand the characteristics of frictional force very well. The wiki says the heat lost during a movement on a frictional surface is f x d... uh, why?

You did include the heat lost due to friction in your energy equation--that's the $u (M+m) g D$ term. The frictional force here is $f_k=u(M+m) g$.

According to Newton's 3rd law, I thought that if the surface doesn't move, then the work that was "supposed to done by an object on the surface" is lost as heat... but what if "the surface," as in the above case, IS moving? Also, if the Earth served as "the surface", wouldn't it acquire a very very small acceleration when an object slides on it?

Why, then. does the energy become heat but not the Earth's kinetic energy?

Any transfer to the Earth's rotational energy from this process would be so small that the distance traveled D would be the same whether you tried to include this effect or not.

 

[aesnix]

Thanks for your help, Alphysicist. I'm still confused, though; this is probably because I didn't think carefully enough while studying, so I guess I should study for a while before I can restate my problem clearly enough. Thanks again!

 

[tiny-tim]

Welcome to PF!

According to Newton's 3rd law, I thought that if the surface doesn't move, then the work that was "supposed to done by an object on the surface" is lost as heat... but what if "the surface," as in the above case, IS moving? Also, if the Earth served as "the surface", wouldn't it acquire a very very small acceleration when an object slides on it?

Why, then. does the energy become heat but not the Earth's kinetic energy?

Hi aesnix! Welcome to PF!

(Actually, Newton's 3rd law is the one about equal and opposite reactions, which doesn't apply to heat! This is just conservation of energy …

http://en.wikipedia.org/wiki/Newton's_third_law#Relationship_to_the_conservation_laws:

Conservation of energy was discovered nearly two centuries after Newton's lifetime, the long delay occurring because of the difficulty in understanding the role of microscopic and invisible forms of energy such as heat and infrared light.

Any energy lost in an inelastic collision is turned into heat, noise, vibration, etc, whether the surface is moving or not.

And in exam questions, you can always assume that the Earth is fixed, unless the question specifies otherwise!

 

[aesnix]

Thanks for your help Tiny-tim.

Still, to avoid some misunderstanding arising from my poor statement, I think I should post the standard solution: (the solution mentioned "internal energy," but I don't want to use this concept because it wasn't taught in class, nor was it mentioned in my text. Well this is what education is like in my country. :( )

mv = (M+m)v'...v' = mv/(M+m)
1/2mv^2 - fd = 1/2(M+m)v'^2
1/2(M+m)v'^2 - u(M+m)gD = 0

This is how I tried to interpret the sol in my way:

I split the whole process into 2 stages-

1. From the instant when the bullet hit the block, the f had been doing negative work on the bullet, with the reaction force -f doing positive work on the block, until the bullet and the block had the same velocity. Since f was a lot larger than u(M+m)g and the stage lasted a very short time, this stage can be treated as an inelastic collision with no external force applying on the system. (I doubt the validity of this assumption) That is,
mv = (M+m)v'...(treat this stage as an inelastic collision with conserved momentum)
1/2mv^2 - fd = 1/2(M+m)v'^2...(a portion fd of the total kinetic energy was lost as heat, with the rest being the total kinetic energy at the end of stage 1)

2. Now the bullet and the block had the same velocity and could be seen as a composite body with mass M+m, speed v', moving on the rough surface. The frictional force from the surface did negative work on the body until it came to a stop. That is,
1/2(M+m)v'^2 - u(M+m)gD = 0...(the total kinetic energy was lost as heat)

Is that reasonable?

 

[tiny-tim]

Hi aesnix!

Since f was a lot larger than u(M+m)g and the stage lasted a very short time, this stage can be treated as an inelastic collision with no external force applying on the system. (I doubt the validity of this assumption)

No, that's fine … momentum is ALWAYS conserved in collisions (whether elastic or not).

Is that reasonable?

Yes, that's very good!

oh … but stop bothering about heat … the question doesn't ask you about heat (and it isn't all heat anyway) … you only need to know the amount of energy lost, not where it went!

(Do you bother about where the weight you lost went? )

 

[aesnix]

No, that's fine … momentum is ALWAYS conserved in collisions (whether elastic or not).

Well what I meant was, can that stage be seen as a collision? More specifically, can I split the process into stages? After all there was indeed a frictional force applied by the surface; that was an external force for my system. Besides, the original question did not state that f was large.

 

[tiny-tim]

Well what I meant was, can that stage be seen as a collision? More specifically, can I split the process into stages? After all there was indeed a frictional force applied by the surface; that was an external force for my system. Besides, the original question did not state that f was large.

Yes … always split the process into stages if there's an inelastic coliision.

Bullets move very fast, so you can regard the friction in the initial stage as being negligible (a question will indicate if it intends otherwise … don't worry about making the questions more difficult than they already are).

 

[aesnix]

Thanks a lot Tiny-tim!