This page supports the multimedia tutorial Projectiles .
How far does a projectile 'fall away' from the line along which it
is aimed?
What happens if you aim at a target, and if the target begins to fall
just as the projectile is launched?
The Monkey and the Hunter is a physics question so old that
it has a heritage listing. As far as we know, it is purely imaginary.
A monkey hangs from a tree. A hunter aims a rifle at him and fires.
At the instant that the gun fires, the monkey lets go of the branch
and begins to fall, thinking that he will thus fall below the trajectory
of the bullet. (Monkeys don't study physics.) What happens?
Before doing the mathematics, let's look at the situation.
The monkey falls and accelerates downwards at g. The bullet
starts off travelling along the aiming line, but it is also
accelerated downwards at g. So we can imagine its motion as
'falling below the aiming line'. At equal times, it will fall
below this line by an amount equal to the distance fallen by
the monkey. At the time when both have the same horizontal
position, they will both have fallen the same distance. This
is not good news for the monkey.
The aiming line or sighting line is the path taken by light,
which is not affected by gravity (or at least not measurably
affected by the Earth's gravity) and so is a straight line
- the black line in this graph. If the projectile is fired
along this line, its initial velocity, v0,
is along that line. If air resistance may be neglected, then
there are no horizontal forces and so the horizontal component
of velocity, vx, is constant. The vertical component, on the
other hand, is steadily decreased by the acceleration due
to gravity. The resulting trajectory is shown at right.
The sequential positions of the bullet shown at right are
equally spaced, 2 units apart. Because the horizontal component
of the velocity vx is constant, then these represent
equally spaced times and the horizontal position x is just
vxt. The vertical lines between the aiming line
and the trajectory show how far the bullet has fallen below
the aiming line. The vertical component of velocity vy has
an initial value vy0, but decreases due to gravity:
vy = vy0 - gt.
Integrating this with respect to time gives us the vertical
position of the projectile:
y = y0 + vy0t - gt2/2.
We can think of the aiming line as the path of an imaginary
projectile that was not affected by gravity. This would
be:
yaim = y0 + vy0t .
The amount by which the projectile has fallen below the
aiming line is
Δy = yaim - y = gt2/2.
Δy is the height of the vertical red lines in the
figure, which increase in the ratio 0, 1, 4, 9, . . . n2.
Now if you look at the algebra above and put v0 =
0, you will see that gt2/2 is the distance
fallen in time t by an obect starting with zero vertical
velocity. So the monkey falls as far below the aiming
line as the bullet does.
No monkeys were hurt in the making of this
clip
More formally, we could use the standard equations for motion in the x
and y direction. Let the bullet start from (x0,y0) with
velocity components vx0 and vy0 at t = 0. If we neglect
air resistance, there is no acceleration in the x direction, so
x = vx0t so t
= x/vx0.
In the y direction, the acceleration is -g,
so
y = vy0t - ½gt2,
which is the equation for a parabola in y(t): a parabola on a displacement-time graph. Combining the two equations above to eliminate t gives
y = x(vy0/vx0) - (½g/vx02)x2,
which is the equation for a parabola in space, in other words the trajectory of the ball. Now v0, the initial velocity, is parallel to the aiming line,
and is the tangent to the trajectory at t = 0, as shown in the diagram.
So we can substitute tan θ = (vy0/vx0)
and get
y = (tan θ)x - (½g/vx02)x2.
The aiming line is just
yaim = tan θ.x
which we could get from the preceding equation by setting g = 0, because
light is not affected by gravity (or at least not very much by the Earth's
relatively weak field). So the vertical lines in the figure are
Δy = yaim - y
= ½(g/vx02)x2
and, using x = vx0t again, we have
Δy = yaim - y
= ½gt2,
which expresses in an equation what we wrote above: the monkey falls as far below the aiming line as the bullet does.
Thanks to Steven Preece, George Hatsidimitris, Gary Keenan and Tamara
Reztsova for high speed videos, flash animations, aiming and monkey
wrangling.
As of June 2005, relativity
was 100 years old. Our contribution is Einstein
Light: relativity in brief... or in detail. It explains
the key ideas in a short multimedia presentation, which
is supported by links to broader and deeper explanations.
Joe's educational pages
