20080930

Physics quiz question: diagonally-launched projectile

Physics 205A Quiz 3, fall semester 2008
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 1/e, Comprehensive Problem 3.74(c)

A ball is kicked off the edge of a cliff with an initial speed of 18.0 m/s, 10.0° above the horizontal. Neglect air resistance. Choose up to be the +y direction. How long would it take for the ball to fall to the ground 22.0 m below?
(A) 1.82 s.
(B) 2.12 s.
(C) 2.24 s.
(D) 2.46 s.

Correct answer (highlight to unhide): (D)

The following quantities are given (or assumed to be known):

(x0 = 0),
(y0 = 0),
(t0 = 0 s),
y = –22.0 m,
ay = –9.80 m/s2,

where the initial horizontal and vertical velocity components of the initial velocity vector are:

v0x = (18.0 m/s)·cos(+10°) = +17.7 m/s,
v0y = (18.0 m/s)·sin(+10°) = +3.13 m/s.

So in the equations for projectile motion, the following quantities are unknown, or are to be explicitly solved for:

x = v0x·t,

vy - v0y = ay·t,

y = (1/2)·(vy + v0yt,

y = v0y·t + (1/2)·ay·(t)2,

vy2 - v0y2 = 2·ay·y.

With the unknown quantity t to be solved for appearing in the fourth equation, with all other quantities given (or assumed to be known), then it becomes a quadratic equation:

y = v0y·t + (1/2)·ay·(t)2,

0 = –y + v0y·t + (1/2)·ay·(t)2,

where the quadratic formula terms are "c" = –(–22.0 m) = +22.0 m; "b" = +3.13 m/s, and "a" = "–4.90 m/s2, resulting in the roots:

t = –1.82 s, +2.46 s,

of which the positive root (response (D)) is the sole realistic answer, given the initial conditions.

(Response (B) is the time t = sqrt(2·y/ay) it would take for the ball to fall to the ground if it were released from rest; response (A) is the time it would take for the ball to fall to the ground if it were thrown at an angle of 10.0° below the horizontal; while response (C) is merely –(y)/ay).

Student responses
Sections 70854, 70855
(A) : 7 students
(B) : 11 students
(C) : 13 students
(D) : 14 students

Success level: 37%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.73

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