I'm a TA in an introductory course of (classical) mechanics at the university. I've been baffled for the couple two hours at this problem, taken from Resnick, Halliday & Krane (Fourth Edition in english; it's the 3rd ed. in spanish). I initially solved the problem, thinking my reasoning was sound, and was surprised when I discovered my (numerical) answer was not the same as the book's.(adsbygoogle = window.adsbygoogle || []).push({});

After more than two hours of checking the reasoning, the algebra, and numerical evaluations, I still find a numerical solution distinct from the one in the book. So I'm wondering whether the book's got it wrong, or whether I'm missing something in the solution. Anyway, here's the problem and my solution.

Problem(translated from spanish)

A block of mass [tex]m_1=1.88[/tex] kg slides along a frictionless table at a speed of [tex]10.3[/tex] m/s. In front of it, and moving in the same direction, is a block of mass [tex]m_2=4.92[/tex] kg, moving at a speed of [tex]3.27[/tex] m/s. A massless spring with a spring constant [tex]k=11.2[/tex] N/cm is attached to the second block, facing the first block, as shown in the figure. When the blocks collide, what is the maximum compression of the spring?

http://img525.imageshack.us/img525/8029/fig.png [Broken]

My solution

I assumed that when the spring is at maximum compression, the two blocks have momentarily the same speed, as if it were a completely inelastic collision. Alternatively, one can also hop in the center-of-mass frame, and it is clear (if we consider them point masses) that at this moment both blocks are at rest (in this frame), thus having the same velocity. (And if two objects have the same velocity in a given inertial reference frame, they'll have the same velocity inanyinertial reference frame - even in SR.)

Then, we can write the conservation of total linear momentum as

[tex]m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_3[/tex]

where [tex]v_1[/tex] and [tex]v_2[/tex] are the initial velocities of the blocks, respectively, and [tex]v_3[/tex] is their common velocity at the exact instant the spring is maximally compressed. We can directly solve this for [tex]v_3[/tex]:

[tex]v_3 = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2}[/tex]

Now, even if the collision is not elastic, the total energy of the system (blocks+spring) must be conserved. Thekineticenergy is not conserved because part of it is being transfered to the spring as potential energy. We know the speeds of the blocks initially and at maximum compression, so we can in principle solve the problem for the maximum compression. The conservation of total energy can be stated in this case as:

[tex]\frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 = \frac{1}{2} (m_1+m_2) v_3^2 + \frac{1}{2} k x^2[/tex]

where [tex]x[/tex] is the maximum compression of the spring. We know all quantities in this equation except for [tex]x[/tex], which is what is asked (we've calculated [tex]v_3[/tex] in terms of known quantities). Therefore, we can solve for [tex]x[/tex]. After some algebra (which I have thoroughly checked, both by doing it myself several times and also by doing it with Maxima), I get:

[tex]x = \left|v_1 - v_2\right| \sqrt{\frac{m_1 m_2}{k(m_1 + m_2)}}[/tex]

A sign ambiguity in the algebraic derivation is produced when we take the square root, so I've taken the absolute value of [tex]v_1-v_2[/tex] because we expect a positive physical answer.

Evaluating the solution with the numbers given in the problem,

[tex]m_1=1.88[/tex] kg

[tex]m_2=4.92[/tex] kg

[tex]v_1=10.3[/tex] m/s

[tex]v_2=3.27[/tex] m/s

[tex]k=1120[/tex] N/m (converted from the given 11.2 N/cm)

I get (and I've also done the numerical computation many times on the computer):

[tex]x \approx 0.245[/tex] m, or some24.5 cm.

However, the solution in the book is35.9 cm! The given answer has enough significant figures to make it unlikely that it was a printing error.

What is even more baffling is that a friend of mine found the same problem in a newer edition of the book, only with different numbers, but my solution still gives the wrong answer in that edition!

Is there an error in my reasoning? Could the book's numerical answer be wrong in both editions?

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# Homework Help: Two blocks and a spring

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