Physics midterm problem: downwards-aimed shotgun

Physics 205A Midterm 1, spring semester 2009
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 3.62(c)

A shotgun fires a large number of pellets downwards, with some pellets traveling very nearly vertically, and others spread out as much as 3.000° from the downwards vertical direction. Assume that the initial speed of these pellets is 180.0 m/s. Ignore air resistance. Find the maximum horizontal displacement that these pellets will hit the ground, as measured from the vertical. Show your work and explain your reasoning.

Solution and grading rubric:

• p:
  Correct. Finds x- and y-components of initial velocity: v_0x = +9.42 m/s, v_0y = –179.8 m/s. May either use quadratic equation or time-eliminated kinematic equation along the y-direction to respectively find the elapsed time or the final v_y when the pellet hits the ground, which can then be used to find the final horizontal position x where the pellet hits the ground.
• r:
  Nearly correct, but includes minor math errors. Typically solves for elapsed time correctly, but inadvertently uses v_0y instead of v_0x in calculating the horizontal distance traveled.
• t:
  Nearly correct, but approach has conceptual errors, and/or major/compounded math errors. At least attempts to solve for elapsed time before substituting into x = v_0x⋅t.
• v:
  Implementation of right ideas, but in an inconsistent, incomplete, or unorganized manner. Still has a methodical approach based on the kinematic equations of motion, or may estimate distance from appealing to trigonometric relations between initial angle, y and/or x, which would only be true if a_y = 0.
• x:
  Implementation of ideas, but credit given for effort rather than merit.
• y:
  Irrelevant discussion/effectively blank.
• z:
  Blank.

Grading distribution:
Sections 30880, 30881
p: 11 students
r: 4 students
t: 2 students
v: 22 students
x: 6 students
y: 1 student
z: 0 students

A sample "p" response (from student 6447):
A sample "t" response (from student 6363), finding the answer in the absence of gravity:
A sample "v" response (from student 1697), purely appealing to trigonometry: