Resource Lesson
Monkey and the Hunter
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The following problem and diagram are provided courtesy of the University of Pennsylvania
lecture notes
on applied two-dimensional motion concepts.
Problem:
A hunter fires a dart gun with a harmless sedative at a monkey hanging from a vine a distance
h
vertically above the dart gun and a distance
R
horizontally away from the dart gun. The hunter aims directly at the monkey and fires, but just as the hunter fires, the monkey, using its incredible spider-monkey sense, realizes what's up and drops from the vine. Does the monkey avoid the dart?
This problem represents an excellent opportunity to examine the independence of the horizontal and vertical motion of a projectile. Remember that the influence of gravity does not dependent on an object's horizontal velocity.
To solve this problem, we must set up simultaneous equations with the parameters
v
and
t
. In order to set up our equations we must also establish which directions we are going to consider to be positive since the dart is rising while the monkey is falling. To this end, we will let the standard "right" be positive along the x-axis and "up" be positive along the y-axis. Finally, the "end of the dart gun" will serve the role of the origin, (x, y) = (0, 0).
Next, let's set up our usual
H | V
charts for both the monkey and the dart and examine our variables.
monkey
H
V
dart
H
V
v
_{H}
= 0
v
_{o}
= 0 m/sec
a = -
g
v
_{H}
=
v
cos(θ)
v
_{o}
=
v
sin(θ)
a = -
g
R = 0
s = (v
_{o}
t + ½a
t
^{2}
) + h
s = h - ½
g
t
^{2}
R = v
_{H}
t
s = v
_{o}
t + ½a
t
^{2}
s =
v
sin(θ) - ½
g
t
^{2}
In order for the dart to hit the monkey, they must intersect vertically directly under the monkey's starting position at some point above the ground. Remember, the monkey does not have any horizontal velocity.
Setting the vertical positions of the monkey and the dart equal to each other yields the following equation:
#1: v sin(θ)t - ½gt
^{2}
= h - ½gt
^{2}
which simplifies to:
#2: v sin(θ)t = h
Next using R = V cos(θ)t we have
#3: v cos(θ)t = R
Dividing the top equation #2 by the bottom equation #3 yields:
#4: tan(θ) = h/R
Since v and t both cancelled, the hunter will always hit the monkey as long as he initially "aims directly at it." The only instance where this outcome would fail would be if the dart's initial horizontal velocity was insufficient to cover the required range.
Use this
video
(download) prepared by the
physics department at Wake Forest
to see this problem in action. Then, afterwards, look at the animated summary that was captured and analyzed in which the ball (dart) is circled in red and the height of the bull's-eye is denoted by a red horizontal line.
animated summary
The above video of the hunter-monkey problem and its animated sequence allowed you to witness first-hand the result predicted by this lesson's mathematical solution.
Bibliography
Davidson College
Physlet: Monkey (Bunny) Gun
University of Pennsylvania
Lecture Notes: Two-Dimensional Motion Concepts Applied
Wake Forest
Demonstration Video
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