Physics midterm problem: upwards bullet embedding in block

Physics 205A Midterm 2, fall semester 2014
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 7.43

"Bullet Block Experiment"
Veritasium
youtu.be/vWVZ6APXM4w

A real-life experiment[*] (illustrated in webcomic format at right[**]) shot a bullet upwards that embedded in a wood block, and the block (with the bullet inside) was free to move upwards to a maximum height of 0.80 m.

An independent analysis[***] assuming a bullet mass of 2.3×10^–3 kg and a block mass of 0.222 kg estimates that the bullet speed just before entering the block was 384 m/s. Ignore friction, drag, and external forces for this brief collision. Determine whether this claim for the bullet's speed is plausible. Show your work and explain your reasoning using properties of collisions, energy (non-)conservation, and momentum conservation.

[*] Derek Muller (Veritasium), "Bullet Block Explained!" youtu.be/BLYoyLcdGPc.
[**] Ben Dickson (Random Perspective Comic), "Problem Solving," randomperspective.com/comic/80/.
[***] John Rowe, "Bullet in the Block Mystery," compadre.org/IVV/docs/Bullet-Block_Recitation_Tutorial.pdf.

Solution and grading rubric:

• p:
  Correct. Momentum is conserved when bullet embeds in block in a perfectly inelastic collision, then as the bullet-embedded block travels upwards, the translational kinetic energy is converted into gravitational potential energy, and determines that the independent analysis claim of the bullet's initial speed is plausible (to two significant digits) or implausible (due to the very small discrepancy in comparing values) by either:
  1. working backwards to determine expected initial speed of bullet, and compares it to the given value of 384 m/s;
  2. works forwards to determine expected maximum height of bullet-embedded block, and compares it to the given value of 0.80 m;
  3. comparing speed (or translational kinetic energy) of bullet-embedded block just after the perfectly inelastic collision, with the expected speed (or translational kinetic energy) of the bullet-embedded block to travel upwards to a maximum height 0.80 m.
• r:
  Nearly correct, but includes minor math errors. May not have squared translational kinetic energy velocity terms, or simple arithmetic errors, but makes a sound argument based on the numerical values resulting from these errors.
• t:
  Nearly correct, but approach has conceptual errors, and/or major/compounded math errors. Certain parameters are misplaced or misidentified, but at least successfully applied momentum conservation to the perfectly inelastic collision of bullet and block, but mechanical energy conservation of upwards-moving bullet-embedded block has multiple issues.
• v:
  Implementation of right ideas, but in an inconsistent, incomplete, or unorganized manner. At least one conservation law is correct (or nearly correct), other is garbled.
• x:
  Implementation of ideas, but credit given for effort rather than merit. No clear attempt at applying conservation laws.
• y:
  Irrelevant discussion/effectively blank.
• z:
  Blank.

Grading distribution:
Sections 70854, 70855, 73320
Exam code: midterm02veR1
p: 26 students
r: 5 students
t: 7 students
v: 16 students
x: 10 students
y: 2 students
z: 0 students

A sample "p" response (from student 9178) comparing the calculated initial speed of the bullet (given the maximum height of the bullet-embedded block) with the given value of 384 m/s:
A sample "p" response (from student 0000) finding the predicted height that the bullet-embedded block would rise up to, comparing it to the observed 0.80 m height:
A sample "p" response (from student 3918) comparing the translational kinetic energy of the bullet-embedded block (just before rising upwards after the collision) with the gravitational potential energy at its highest point:
A sample "p" response (from student 7007) comparing the speed of the bullet-embedded block (just before rising upwards after the collision) with the speed required to make it up to a height of 0.80 m: