# The Monkey and the Hunter

The curves correspond to the trajectories of darts fired at different speeds. Monkeys and darts of the same colour correspond to their positions at the same moment in time.
In the SVG file, hover over a monkey or dart to highlight those contemporaneous with it. Note that monkey and darts remain in a line parallel to the line connecting their initial positions.

"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 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?

## Discussion

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 the problem was stated in a reference frame in which the Earth is motionless. However, 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 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 the result is still 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).

## Derivation

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

${\displaystyle Y_{\rm {dart}}=V_{Y0}t-{\frac {1}{2}}gt^{2}}$

and

${\displaystyle Y_{\rm {monkey}}=h-{\frac {1}{2}}gt^{2}.}$

They will collide when those altitudes are the same, that is

${\displaystyle V_{Y0}t-{\frac {1}{2}}gt^{2}=h-{\frac {1}{2}}gt^{2}.}$

The term gt² /2 is both present on both sides of the equation, which then can be simplified to

${\displaystyle V_{Y0}t=h.}$

In order for the dart to collide with the monkey, the horizontal positions have to coincide simultaneously with the vertical positions. Using VX0 to denote the initial horizontal velocity of the dart and d to denote the initial horizontal distance to the monkey, the equations of horizontal motion are

${\displaystyle X_{\rm {dart}}=V_{X0}t}$

and

${\displaystyle X_{\rm {monkey}}=d.}$

Equating the horizontal positions gives

${\displaystyle V_{X0}t=d.}$

Since t is the same in both the horizontal and vertical equations, it can be eliminated by

${\displaystyle t={\frac {h}{V_{Y0}}}={\frac {d}{V_{X0}}}\implies {\frac {V_{X0}}{V_{Y0}}}={\frac {d}{h}}.}$

From this we can conclude that the dart will hit the monkey if the initial velocity vector is parallel to the displacement vector from the gun to the monkey--i.e., the gun is pointed at the monkey. To handle the case where h is zero (and avoid dividing by zero), note that the previous equations imply that VY0 must be zero as well; if the monkey starts at the same height as the gun, then a horizontal shot will hit it.