The Monkey and the Hunter
Encyclopedia
"The Monkey and the Hunter" is a thought experiment
often used to illustrate the effect of gravity on projectile motion
.
The essentials of the problem are stated in many introductory guides to physics
, such as Caltech's The Mechanical Universe
television series and Gonick
and Huffman's Cartoon Guide to Physics. In essence, the problem is as follows: A hunter with a blowgun goes out in the woods to hunt for monkeys and sees one hanging in a tree, at the same level as the hunter's head. The monkey, we suppose, releases its grip the instant the hunter fires his blowgun. Where should the hunter aim and when should he fire in order to hit the monkey?
To answer this question, recall that according to Galileo's law, all objects near the Earth's surface fall with the same constant acceleration
, 9.8 metres per second per second (32 feet per second per second), regardless of the object's weight. Furthermore, horizontal motions and vertical motions are independent: gravity acts only upon an object's vertical velocity
, not upon its velocity in the horizontal direction. (This can easily be treated by representing velocity and acceleration as vectors in a Cartesian coordinate system.) The hunter's dart, therefore, falls with the same acceleration as the monkey.
Assume for the moment that gravity was not at work. In that case, the dart would proceed in a straight-line trajectory at a constant speed (Newton's first law). Gravity causes the dart to fall away from this straight-line path, making a trajectory that is in fact a parabola
. Now, consider what happens if the hunter aims directly at the monkey, and the monkey releases his grip the instant the hunter fires. Because the force of gravity accelerates the dart and the monkey equally, they fall the same distance in the same time: the monkey falls from the tree branch, and the dart falls the same distance from the straight-line path it would have taken in the absence of gravity. Therefore, the dart will always hit the monkey, no matter the initial speed of the dart.
Another way of looking at the problem is by a transformation of the reference frame
. Earlier, we stated the problem in a reference frame in which the Earth is motionless. Now, we know that for very small distances on the surface of Earth the acceleration due to gravity can be considered constant to good approximation. Therefore, the same acceleration g acts upon both the dart and the monkey throughout the fall. Transform the reference frame to one that is accelerated upward by the amount g with respect to the Earth's reference frame (which is to say the acceleration of the new frame with respect to the Earth is –g). Because of Galilean equivalence, the (approximately) constant gravitational field (approximately) disappears, leaving us with only the horizontal velocity of both the dart and the monkey.
In this reference frame it is obvious that the hunter should aim straight at the monkey, since the monkey is stationary. Since angles are invariant under transformations of reference frames, transforming back to the Earth's reference frame we still get the result that the hunter should aim straight at the monkey. While this approach has the advantage of making the results intuitively obvious, it suffers from the slight logical blemish that the laws of classical mechanics are not postulated within the theory to be invariant under transformations to non-inertial (accelerated) reference frames (see also principle of relativity
).
To write equations for the motion of the monkey and the hunter's dart, use g to denote the acceleration of gravity, t for elapsed time, and h for the initial height of the monkey. Using VY0 to denote the initial vertical speed of the dart, the equations for the vertical motion (altitude) of the dart and the monkey are respectively
and
They will collide when those altitudes are the same, that is
Thought experiment
A thought experiment or Gedankenexperiment considers some hypothesis, theory, or principle for the purpose of thinking through its consequences...
often used to illustrate the effect of gravity on projectile motion
Projectile motion
The motion in which a body is thrown or projected is called Projectile motion.The path followed by a projectile is called its trajectory, which is directly influenced by gravity....
.
The essentials of the problem are stated in many introductory guides to physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, such as Caltech's The Mechanical Universe
The Mechanical Universe
The Mechanical Universe... And Beyond, is a 52-part telecourse filmed at the California Institute of Technology, and produced by Caltech and INTELECOM Intelligent Telecommunications . The series introduces university level physics, covering topics from Copernicus to quantum mechanics...
television series and Gonick
Larry Gonick
Larry Gonick is a cartoonist best known for The Cartoon History of the Universe, a history of the world in comic book form, which he has been publishing in installments since 1977...
and Huffman's Cartoon Guide to Physics. In essence, the problem is as follows: A hunter with a blowgun goes out in the woods to hunt for monkeys and sees one hanging in a tree, at the same level as the hunter's head. The monkey, we suppose, releases its grip the instant the hunter fires his blowgun. Where should the hunter aim and when should he fire in order to hit the monkey?
To answer this question, recall that according to Galileo's law, all objects near the Earth's surface fall with the same constant acceleration
Acceleration
In physics, acceleration is the rate of change of velocity with time. In one dimension, acceleration is the rate at which something speeds up or slows down. However, since velocity is a vector, acceleration describes the rate of change of both the magnitude and the direction of velocity. ...
, 9.8 metres per second per second (32 feet per second per second), regardless of the object's weight. Furthermore, horizontal motions and vertical motions are independent: gravity acts only upon an object's vertical velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
, not upon its velocity in the horizontal direction. (This can easily be treated by representing velocity and acceleration as vectors in a Cartesian coordinate system.) The hunter's dart, therefore, falls with the same acceleration as the monkey.
Assume for the moment that gravity was not at work. In that case, the dart would proceed in a straight-line trajectory at a constant speed (Newton's first law). Gravity causes the dart to fall away from this straight-line path, making a trajectory that is in fact a parabola
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
. Now, consider what happens if the hunter aims directly at the monkey, and the monkey releases his grip the instant the hunter fires. Because the force of gravity accelerates the dart and the monkey equally, they fall the same distance in the same time: the monkey falls from the tree branch, and the dart falls the same distance from the straight-line path it would have taken in the absence of gravity. Therefore, the dart will always hit the monkey, no matter the initial speed of the dart.
Another way of looking at the problem is by a transformation of the reference frame
Reference frame
Reference frame may refer to:*Frame of reference, in physics*Reference frame , frames of a compressed video that are used to define future frames...
. Earlier, we stated the problem in a reference frame in which the Earth is motionless. Now, we know that for very small distances on the surface of Earth the acceleration due to gravity can be considered constant to good approximation. Therefore, the same acceleration g acts upon both the dart and the monkey throughout the fall. Transform the reference frame to one that is accelerated upward by the amount g with respect to the Earth's reference frame (which is to say the acceleration of the new frame with respect to the Earth is –g). Because of Galilean equivalence, the (approximately) constant gravitational field (approximately) disappears, leaving us with only the horizontal velocity of both the dart and the monkey.
In this reference frame it is obvious that the hunter should aim straight at the monkey, since the monkey is stationary. Since angles are invariant under transformations of reference frames, transforming back to the Earth's reference frame we still get the result that the hunter should aim straight at the monkey. While this approach has the advantage of making the results intuitively obvious, it suffers from the slight logical blemish that the laws of classical mechanics are not postulated within the theory to be invariant under transformations to non-inertial (accelerated) reference frames (see also principle of relativity
Principle of relativity
In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference....
).
To write equations for the motion of the monkey and the hunter's dart, use g to denote the acceleration of gravity, t for elapsed time, and h for the initial height of the monkey. Using VY0 to denote the initial vertical speed of the dart, the equations for the vertical motion (altitude) of the dart and the monkey are respectively
and
They will collide when those altitudes are the same, that is
-
The term gt² /2 is both present on both sides of the equation, which then can be simplified to
Given a non-zero
it can be rewritten to define when that occurs:
And given a non-zero
the only possible values that satisfies the equation are h = 0 and any value of t. In short, there is always a time t when both the dart and the monkey will collide vertically.
That said, this thought experimentThought experimentA thought experiment or Gedankenexperiment considers some hypothesis, theory, or principle for the purpose of thinking through its consequences...
has been criticized as completely out of touch with reality. According to the conditions as explained, it would be impossible to hit the monkey if it were tethered to the tree! However, real world blowgun hunters either take into account the effect of gravity when targeting a fix target or miss the shot; as a consequence, the falling monkey will safely get away.
External links
- http://www.waowen.screaming.net/revision/force&motion/mandh.htm
- http://www.ap.stmarys.ca/demos/content/mechanics/monkey_and_hunter/monkey_and_hunter.htm
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