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):

(

*x*

_{0}= 0),

(

*y*

_{0}= 0),

(

*t*

_{0}= 0 s),

*y*= –22.0 m,

*a*= –9.80 m/s

_{y}^{2},

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

*v*

_{0x}= (18.0 m/s)·cos(+10°) = +17.7 m/s,

*v*

_{0y}= (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*=

*v*

_{0x}·

*t*,

*v*-

_{y}*v*

_{0y}=

*a*·

_{y}*t*,

*y*= (1/2)·(

*v*+

_{y}*v*

_{0y})·

*t*,

*y*=

*v*

_{0y}·

*t*+ (1/2)·

*a*·(

_{y}*t*)

^{2},

*v*

_{y}^{2}-

*v*

_{0y}

^{2}= 2·

*a*·

_{y}*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*=

*v*

_{0y}·

*t*+ (1/2)·

*a*·(

_{y}*t*)

^{2},

0 = –

*y*+

*v*

_{0y}·

*t*+ (1/2)·

*a*·(

_{y}*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/s

^{2}, 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*/

*a*) 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°

_{y}*below*the horizontal; while response (C) is merely –(

*y*)/

*a*).

_{y}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