Inelastic Collisions and Dissipation of Energy

[chiurox]

Homework Statement

Block A of mass, mA = 2.0 kg, is moving on a frictionless surface with a velocity of 5.0 m/s to the right and another block B of mass, mB = 8.0 kg, is moving with a velocity of 3.0 m/s to the left, as shown in the diagram below. The two block eventually collide.

d. If the initial velocity of Block A is 4.0 m/s to the right and -30 J is dissipated in the collision, what was the initial velocity of Block B?

The Attempt at a Solution

mavai + mbvbi = (ma+mb) vf
(0.5)mavai² + (0.5)mbvbi² = Ei
(0.5)(ma+mb)vaf² = Ef
Ef – Ei = -30 J

(4.0) + (4.0)vbi = (5.0) vf
(16) + (4.0)vbi² = Ei
(5.0)vf² = Ef
Ef – Ei = -30 J

I don't know how to go from here. Any help?

 

[catkin]

I don't know where to go from here, either. The question as you have given it gives two velocities for blocks A and B -- and you've made your query harder to understand by missing out the formulae you are using and poor formatting of your attempt.

 

[chiurox]

I don't know where to go from here, either. The question as you have given it gives two velocities for blocks A and B -- and you've made your query harder to understand by missing out the formulae you are using and poor formatting of your attempt.

About the poor formating, yes, I noticed it's pretty bad. I pasted directly from Word because I'm still not familiar with the formating thing we have for this forum.

 

[dotman]

Hello,

Momentum is conserved, as you have shown. This can be used to show how the before and after kinetic energies are related-- see

http://hyperphysics.phy-astr.gsu.edu/hbase/inecol.html"

I think this may help a bit. If you take a look at the formulas you have written already:

E_f - E_i = -30

m_av_{ai} + m_bv_{bi} = (m_a + m_b)v_f

\frac{1}{2}m_a{v_{ai}}^2 + \frac{1}{2}m_b{v_{bi}}^2 = E_i

\frac{1}{2}(m_a + m_b){v_f}^2 = E_f

You have four equations and four unknowns (E_i , E_f, v_{bi}, v_f) so you should be able to solve this. I would substitute my expressions for E_i and E_f into the first equation, and then use the second equation to substitute in for either v_f or v_{bi}.

Finally, check your interpretation of the first equation. You stated that "-30J are dissipated"-- does this mean that 30J are actually gained in the collision? Or were they meaning that 30J were dissipated (lost)? If the former, your equation has a sign error. If the latter (which is most likely), you should be ok.

Solve it with the letters, and put the numbers in last.

 

[catkin]

Block A of mass, mA = 2.0 kg, is moving on a frictionless surface with a velocity of 5.0 m/s to the right

[snip]
d. If the initial velocity of Block A is 4.0 m/s to the right

[snip]

No amount of conservation of momentum and energy equations is going to solve the riddle of which velocity to use! Has something significant been lost along with parts a, b and c of the question?

 

[chiurox]

Alright, I solved it.
I ended up isolating just the Vf² and the Vf into one equation so I just solved it with the quadratic formula. So I got two final velocities, one positive and one negative. Since the initial velocity of block B has to be negative, Vf had to be negative too.

By dissipated I mean the energy is lost, that's why I put negative.
So the initial velocity for Block B is -1.4 m/s.
I guess it makes sense, could you guys confirm?

Thank you.

 

[dotman]

Alright, I solved it.
I ended up isolating just the Vf² and the Vf into one equation so I just solved it with the quadratic formula. So I got two final velocities, one positive and one negative. Since the initial velocity of block B has to be negative, Vf had to be negative too.

By dissipated I mean the energy is lost, that's why I put negative.
So the initial velocity for Block B is -1.4 m/s.
I guess it makes sense, could you guys confirm?

Thank you.

Hello,

That's not what I got, but I just did the problem real quick on a napkin, and may have made a mistake. What value did you get for Vf? You can at least check if your answer "makes sense".

More importantly, why does the initial velocity of block B have to be negative? And even if it is, this does not imply that Vf is negative, necessarily-- what if Vb is very very slightly negative? (What if its zero?)

Lastly, there is a simpler way to do this problem. Did you check out that link?

EDIT: Actually, re-looking at that link, it looks like its only valid for the case of Block B being motionless. So that's no good for here, necessarily.

 

[chiurox]

Hello,

That's not what I got, but I just did the problem real quick on a napkin, and may have made a mistake. What value did you get for Vf? You can at least check if your answer "makes sense".

More importantly, why does the initial velocity of block B have to be negative? And even if it is, this does not imply that Vf is negative, necessarily-- what if Vb is very very slightly negative? (What if its zero?)

Lastly, there is a simpler way to do this problem. Did you check out that link?

EDIT: Actually, re-looking at that link, it looks like its only valid for the case of Block B being motionless. So that's no good for here, necessarily.

Yeah, I checked that link and saw that one of the masses was motionless, so the situation is a bit different from this one.
I redid the problem and got two answers: 0.9 m/s or -13 m/s for the initial velocity of block B and two final velocities for both objects as 1.5 m/s or -9.5 It makes sense I guess, because block A is traveling about twice as faster and hits the bigger block B making Block B travel a bit faster while block A has energy dissipated due to the collision.
Is that what you got?

 

[dotman]

I redid the problem and got two answers: 0.9 m/s or -13 m/s for the initial velocity of block B and two final velocities for both objects as 1.5 m/s or -9.5 It makes sense I guess, because block A is traveling about twice as faster and hits the bigger block B making Block B travel a bit faster while block A has energy dissipated due to the collision.
Is that what you got?

Not quite what I got. From a physical standpoint, you're right, the answers kind of make sense. In the first case, the big block is moving slowly to the right, and the small block comes up and hits it, and the two now move slightly faster than Block B, as expected. In the second case, the massive block flies in fast, to the left, strikes block A, reverses it, and they continue left more slowly. That's all well and good.

You can check, however, to see if your answers are correct or not. They have to satisfy your initial equations-- conservation of momentum, and your energy equation. Let's have a look:

First, vb = 0.9 m/s and vf = 1.5 m/s:

m_av_{ai} + m_bv_{bi} = (m_a + m_b)v_f

\ \Rightarrow (2){(4)} + (8){(0.9)} = (2 + 8){(1.5)}

\ \Rightarrow 8 + 7.2 = 15

\ \Rightarrow 15.2 = 15

So that's good, the slight difference is due to rounding. Now for the energy equation:

\frac{1}{2}m_a{v_{ai}}^2 + \frac{1}{2}m_b{v_{bi}}^2 = E_i

\frac{1}{2}(m_a + m_b){v_f}^2 = E_f

E_i - E_f = 30

\ \Rightarrow \frac{1}{2}m_a{v_{ai}}^2 + \frac{1}{2}m_b{v_{bi}}^2 - [ \frac{1}{2}(m_a + m_b){v_f}^2] = 30


\ \Rightarrow \frac{1}{2}(2){(4)}^2 + \frac{1}{2}(8){(0.9)}^2 - \frac{1}{2}(2 + 8){(1.5)}^2 = 30

\ \Rightarrow 16 + 3.24 - 11.25 = 30

\ \Rightarrow 19.24 - 11.25 = 30

\ \Rightarrow 7.99 \not= 30 So there's a problem here.

You can do the same kind of thing for your other set of answers, to see what fits and what doesn't.