# Homework Help: Find average velocity of a sphere which expands and moves

1. Apr 22, 2015

### Quarlep

1. The problem statement, all variables and given/known data
Find the average velocity of shell while its moves a velocity v' and expands a velocity v.
Expansion velocity v
Movement velocity v'

2. Relevant equations
I think theres no need an equation.

3. The attempt at a solution
I try to find average velocity but its vector.So I made max velocity v+v' and min velocity v-v' then I add them and divide two and I find v but I think I made a mistake.

2. Apr 22, 2015

### Puma

That's right. If you draw a line in any direction through the shell's centre those expansion velocities cancel out so you are just left with v

3. Apr 22, 2015

### Quarlep

Expansion velocities cant cancel v is expansion velocity v' is object velocity.I didnt mention it but shell is a surface of sphere.And I didnt understand your prove

4. Apr 22, 2015

### Puma

Ok ignore v' eg imagine the sphere is stationary what is the average velocity of the sphere shell. The speed is v but the velocity which is a vector cancels out: if you imagine any point on the sphere - there is one going in exactly the opposite direction so when you add them all up the answer is 0.

5. Apr 22, 2015

### Quarlep

So that expansion velocities cancels and left only v which speed of sphere

6. Apr 22, 2015

### Quarlep

I am so so sorry .I made a mistake in my equation it must be v'-v not v-v' my fault.
I want to ask another question.Before sphere moves, shell energy will be 1/2mv^2(v is shell expansion speed) but after sphere moves shell energy will be 1/2mv^2 (v is the sphere velocity) isnt it ?

7. Apr 23, 2015

### AlephNumbers

The expansion velocities don't "cancel out" unless we are considering the velocity of the center of mass of the sphere. In which case you are right, the velocity of the center of mass of the sphere should stay constant.

No, I don't believe so. Energy is a scalar quantity, so the expansion velocities will not conveniently cancel out. The only way that I am able to come to any sort of meaningful conclusion regarding the kinetic energy of the sphere is by considering the kinetic energy of the sphere at the very instant that the sphere begins to expand. Here is a picture to help you visualize the situation.

8. Apr 23, 2015

### Quarlep

I am not talking about kinetic energy of sphere.I am talking about shell energy

9. Apr 23, 2015

### AlephNumbers

Sorry, the shell. I accidentally used the term sphere instead of shell. Please forgive me.

10. Apr 23, 2015

### Quarlep

When shell is not moving shell energy will be the "energy " which that energy causes expension of shell and thats the 1/2m2v2.Then shell moves and the speed of shell will be (as you mentioned vectors of speeds which you add them and calculations shows a v which is the speed of movement) so energy of shell will be 1/2mshellv2 v is speed of sphere

Last edited: Apr 23, 2015
11. Apr 23, 2015

### AlephNumbers

My good chap, I am afraid that any solution to your somewhat vague question is significantly more complicated. Why don't you examine the problem further, and maybe sleep on it. Then show us what you have come up with. Perhaps my diagram will give you some inspiration.

12. Apr 23, 2015

### Quarlep

Look the prove of speed.

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13. Apr 23, 2015

### haruspex

Yes, that correctly shows the expansion cancels out when computing average velocity. But average KE is not given by squaring average velocity. You have to find the KE of each element separately and then integrate.
There's an analogous result in statistics: the variance is the mean square minus the square of the mean.

14. Apr 24, 2015

### Quarlep

I didnt understand this and here what I found

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15. Apr 24, 2015

### haruspex

The integrals on the left are ok, but they don't give the answer on the right. please post your steps.

16. Apr 24, 2015

### Quarlep

Here the steps

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17. Apr 24, 2015

### haruspex

The integrals of sin2 and cos2 cannot cancel to zero. The integrands are non-negative everywhere.
You'll find it simpler if you collapse v2sin2 + v2 cos2 to v2.

18. Apr 24, 2015

### Quarlep

they can cancel to zero.If we integrate (sinx)2 between 0 and π we get π/2 then we integrate same thing again between π and 0 we get -π/2 theres nothing wrong look carefully.Only my first equation can be wrong.But you said thats true so we get mv2

19. Apr 24, 2015

### haruspex

There's your problem, you're running the integral backwards, getting a negative result from an integrand that's never negative.

20. Apr 24, 2015

### Quarlep

You told me the right side of the equation is true.If its true then we need to integrate backward.If you think different way prove me using math.Or my first equation is wrong.