The following problem and diagram are provided courtesy of the University of Pennsylvania lecture notes on applied twodimensional 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 spidermonkey 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 xaxis and "up" be positive along the yaxis. 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 + ½at^{2}) + h
s = h  ½gt^{2} 
R = v_{H}t

s = v_{o}t + ½at^{2}
s = v sin(θ)  ½gt^{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'seye is denoted by a red horizontal line.
The above video of the huntermonkey problem and its animated sequence allowed you to witness firsthand the result predicted by this lesson's mathematical solution.
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