Physics midterm problem: football slingshot launch angle

Physics 205A Midterm 1, fall semester 2015
Cuesta College, San Luis Obispo, CA

"Winning The FIFA World Cup With The Slingshot"
The Slingshot Channel
youtu.be/Om0O0_gr6Vo

Slingshot innovator Jörg Sprave constructed a "football crossbow" to launch a ball at an upwards angle above the horizontal, such that its trajectory lands on the other side of a 100 m field, 2.8 seconds afterwards.[*] Determine whether the launch angle was less than or greater than 45° above the horizontal. Assume that the starting and ending heights of the ball are both at ground level. Neglect air resistance. Show your work and explain your reasoning using properties of projectile motion.

[*] Jörg Sprave, "Winning The FIFA World Cup With The Slingshot," youtu.be/Om0O0_gr6Vo.

Solution and grading rubric:

• p:
  Correct. Given elapsed time (t = 2.8 s) and final horizontal (x = +100 m) and vertical position (y = 0 m), finds horizontal v_0x and vertical v_0y components of the initial velocity vector, and either calculates the launch angle θ = 21° (or deduces that θ must be less than 45° because v_0y is less than v_0x), or uses actual v_0x = +36 m/s and sets θ = 45° to find a presumptive v_0y = +36 m/s, then argues that the ball with this (presumptive) launch angle and initial vertical velocity component would still be in the air at t = 2.8 s and thus the initial velocity component v_0y must be less than +36 m/s, making θ less than 45°.
• r:
  Nearly correct, but includes minor math errors. At least successfully solves for the horizontal v_0x and vertical v_0y components of the initial velocity vector, but calculation and/or conclusion from finding/deducing launch angle θ is garbled.
• t:
  Nearly correct, but approach has conceptual errors, and/or major/compounded math errors. At least enough steps are shown that would theoretically result in a complete answer, multiple errors notwithstanding. May have incorrectly solved for the vertical component of the initial velocity vector v_0y by erroneously setting v_y = 0 at t = 2.8 s (when v_y = 0 momentarily at its highest point at t = 1.4 s).
• v:
  Implementation of right ideas, but in an inconsistent, incomplete, or unorganized manner.
• x:
  Implementation of ideas, but credit given for effort rather than merit.
• y:
  Irrelevant discussion/effectively blank.
• z:
  Blank.

Grading distribution:
Sections 70854, 70855, 73320
Exam code: midterm01j0r6
p: 30 students
r: 14 students
t: 7 students
v: 12 students
x: 14 students
y: 1 student
z: 0 students

A sample "p" response (from student 2681), explicitly solving for the launch angle:
A sample "p" response (from student 8420), arguing that the launch angle must be less than 45° as v_0x > v_0y:
A sample "p" response (from student 1959) presuming that v_0y = v_0x, and finds that since the ball would be in the air longer than 2.8 s, then v_0y must be less than v_0x, making the launch angle less than 45°: