Physics midterm problem: jumping frog

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

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 3.55(b)

[20 points.] A frog jumps at an angle of 40° from the horizontal, and lands back on the level ground 0.87 s later, a distance of 4.4 m away. Solve for the x- and y-components of its initial velocity. Show your work and explain your reasoning.

Solution and grading rubric:

• p = 20/20:
  Correct. Solves for average horizontal speed from given horizontal displacement and elapsed time, this must also be the initial horizontal velocity component v_ix. Then can use the initial trajectory angle to solve for v_iy = v_ix·tan(40°). Or can use half of flight time, and v_fy = 0 at top of trajectory to find v_iy .
  
• r = 16/20:
  Nearly correct, but includes minor math errors. May have shifted a decimal place, or used tan(40°) = adjacent/opposite, or apparent miscalculation of otherwise correct equations.
  
• t = 12/20:
  Nearly correct, but approach has conceptual errors, and/or major/compounded math errors. At least has v_ix = ∆x/∆t. Garbles attempt at using kinematics to find v_iy (typically using 0.87 s as the time to travel to its highest height).
  
• v = 8/20:
  Implementation of right ideas, but in an inconsistent, incomplete, or unorganized manner. Still has a methodical approach based on kinematics and/or trigonometry. May claim that ∆x/∆t is the magnitude of the initial velocity vector, which is then broken into v_ix and v_iy components using cos(40°) and sin(40°).
  
• x = 4/20:
  Implementation of ideas, but credit given for effort rather than merit.
  
• y = 2/20:
  Irrelevant discussion/effectively blank.
  
• z = 0/20:
  Blank.

Grading distribution:
Sections 70854, 70855, 72177
p: 26 students
r: 9 students
t: 9 students
v: 9 students
x: 10 students
y: 2 students
z: 0 students

A sample of a "p" response, using the time of flight and initial velocity vector angle to solve for the magnitude of the initial velocity vector, which is then broken into x- and y- components (from student 2323):
Another "p" response, using the time of flight and the horizontal range to solve for the x-component of the initial velocity vector, and together with the initial velocity vector direction is used to find the y-component of the initial velocity vector (from student 6405):
An "r" response, but with the entire flight time used to find the y-component of the initial velocity vector (from student 1251):
Another "r" response, with a misapplication of the tangent function (from student 2051):
A sample of a "v" response, where the x-component of the initial velocity vector is used as the magnitude of the initial velocity vector (from student 9685):