The Monkey and the Hunter
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
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 zero the only possible values that satisfy 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.