# Motion of particles

1. Jun 3, 2010

### thereddevils

1. The problem statement, all variables and given/known data

Particle A with mass m1 and particle B with mass m2 move with speed u1 and u2 respectively towards one another on a smooth horizontal plane . THe coefficient of restitution between particle A and particle B is e . If m1>em2 , and particle A continues to move without changing its direction after colliding with particle B, show that

$$u_1>\frac{(1+e)m_2u_2}{m_1-em_2}$$

2. Relevant equations

3. The attempt at a solution

some hints?

2. Jun 3, 2010

### kuruman

Hint 1: What equation is always relevant when you have a collision?
Hint 2: What is the definition of the coefficient of restitution?

3. Jun 4, 2010

### thereddevils

thanks ,

m1u1+m2u2=m1v1+m2v2 since its an elastic collision .

From newton's law of restitution ,

v2-u2=e(v1-u1)

do i play around with this two formulas ?

4. Jun 4, 2010

### kuruman

Whether the collision is elastic or inelastic, momentum is conserved. In fact if the coefficient of restitution is anything but 1, the collision is inelastic, i.e. kinetic energy is not conserved.
Not quite. You need absolute values for the relative velocity.
|v2-u2| = e|v1-u1|
Yes, but you have to use correctly the information that the two objects are moving in opposite directions before the collision but in the same direction after the collision. Be mindful of where you put plus and minus signs in front of each velocity. Note that the coefficient of restitution is essentially the ratio of relative speeds.

5. Jun 4, 2010

### thereddevils

thanks for explaining,

so i think the first equation should be

$$m_1u_1-m_2u_2=m_1v_1+m_2v_2$$

I am not sure with this restitution part. What exactly is e ? epsilon?

I only know that Newton's law of restitution which states that

$$v_2-u_2=e(v_1-u_1)$$

If i include the modulus, i don find a way to solve the equations except if i square them, do I ?

6. Jun 4, 2010

### kuruman

Your momentum conservation equation is correct with the understanding that v1, v2, u1 and u2 represent speeds, i.e. they are all positive quantities.

The coefficient of restitution is e, given by the equation

$$e=\frac{|v_2-u_2|}{|v_1-u_1|}$$

You don't need to square anything. As I said earlier, think of the numerator and denominator as the relative speeds. Speed is always positive.

In this case, before the collision the particles are moving in opposite directions, therefore the relative speed is the sum of the speeds of the two masses. That's a positive number. After the collision, the particles are moving in the same direction, which means that the relative speed is the difference of the speeds of each mass. That's also a positive number if you subtract the smaller speed from the larger one.

7. Jun 5, 2010

### thereddevils

is the equation of restitution

$$e=\frac{V_2+U_2}{U_1-V_1}$$ ?

8. Jun 5, 2010

### kuruman

I am sorry, I misread the problem. I thought subscripts "1" and "2" referred to "before" and "after" the collision. Actually they label the particles. So if "u" stands for "before" and "v" stands for "after", the correct expression for the coefficient of restitution is

$$e=\frac{|v_2-v_1|}{|u_2-u_1|}$$

Start from there and, again, sorry about the confusion.

9. Jun 5, 2010

### thereddevils

oh its ok , so the formula is now

$$e=\frac{v_1-v_2}{u_1+u_2}$$

I tried playing around with the equations but i don see a way to get rid of v_1 and v_2
since they are not required in the final answer .

10. Jun 5, 2010

### kuruman

Note that the problem asks you to show that u1 is greater than a certain expression if the mass that has speed u1 is to continue moving in the same direction after the collision.

Which way do you think the mass will move if it has speed less than that expression?
What about if it has speed equal to that expression?

Answer these two questions and you will see what to do with v1 and v2.

11. Jun 5, 2010

### thereddevils

But that tells me the direction of these particles , i still do not know their magnitudes .

12. Jun 6, 2010

### kuruman

Think it through. If you can find what u1 must be so that particle 1 is at rest after the collision, then any value of u1 greater than that will give you the inequality that you want. Now, if particle 1 is at rest after the collision, you know v1, do you not? So go ahead and find the threshold value of u1.

13. Jun 6, 2010

### thereddevils

ok .

using the first formula , m1u1-m2u2=m1v1+m2v2

v1 is 0 ,

$$u_1=\frac{m_2v_2+m_2u_2}{m_1}$$ ---1

then , from formula 2, put v1=0 too , v2+u2=eu1

put this into 1 , i got u1=(m2 e u2)/(m1)

nvm , this is definitely wrong but is this what you meant .

14. Jun 6, 2010

### kuruman

Correct.

Incorrect. Try solving for eu1 again.

15. Jun 6, 2010

### thereddevils

thanks Kuruman , i got it ...... finally !