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Motion in Two Dimensions

We will explore the use of the equations for motion for more than one dimension by explicit application. The traditional starting point is with free-fall motion since historically, this is how Galileo started the chain of studies that eventually led Newton to develop his three laws of motion.

Galileo was the founder of modern physics not merely because of the ideas he introduced and studied, but mostly because of the way he thought about them. In particular, he is the first famous practioner of the gedanken or thought experiment. You imagine a situation in which your experience (what physicists call physical intuition) tells you what should happen under certain conditions. If in the course of imagining what will happen you find a contradiction between what you expect and what your understanding of the principles of physics would allow, then you know that your understanding is limited or that the principle needs to be questioned. The advantage of gendanken experiments is that you can imagine situations which would be difficult or practically impossible to achieve in a laboratory setting. Nonetheless physicists use gedanken experiments even today to test their notions of how objects which have never been directly seen, black holes for example, must behave according to physics principles. If the behavior of the system due to a certain principle of physics leads to a contradiction with another principle, the deeper principle is usually held fixed in the mind while physicists explore the "weakness" in the the contradicting principle. In this way researchers can choose which problems are likely to yield significant insights.

Galileo's thought experiment is shown in the picture below.

In the picture one observer on a boat moving with speed v with respect to the shore sees a ball fall from the top of the mainsail mast. The observer on shore also watches the ball fall. Do the two observers see the same thing for the motion? They cannot. The observer on the boat moves horizontally with the same speed as the horizontal motion of the ball (since both were attached to the boat and no other forces in the horizontal direction are acting once the ball is released). Therefore the observer on the boat sees a trajectory or flight path which is straight down. The observer on the shore, however, sees the trajectory shown below

The shape of the trajectory is parabolic. How do we know? We have to have some idea of what the shape should look like. From this thought experiment Galileo concluded that the trajectory results from the independence of motion along the horizontal and vertical directions. The simplest idea is to say that the vertical motion is unaffected by the horizontal motion of the boat. Therefore, the difference in the two trajectories seen by the observers results only because they see the ball with different relative horizontal velocities. The observer on the boat sees a relative horizontal velocity of zero for the whole time the ball falls since gravity only accelerates things in the vertical direction. The shore observer sees this vertical motion AND the relative horizontal velocity of the boat and ball. We can use Maple to tell us if this idea gives us the right behavior. Assuming no change in the form of the equations of motion, the shore observer should see the initial velocities and positions for x (horizontal) and y (vertical) used in the appropriate equations for motion as follows:

x motion	y motion
x = x0 + v0xt	y = y0 + v0yt - ½ gt2

We can now explore why nature is so kind to us as to allow us to assume independence of the vertical and horizontal axes, i.e. motion along the vertical axis is unaffected by motion along the x axis and vice versa! This really simplifies life because it says we can just apply the same equations of motion we are used to using for the vertical direction and the horizontal direction without any worries as to how one affects the other.

As an example, you can hold down the right mouse button on the picture to the left and select play from the popup menu (if you are using Netscape) to see the vectors associated with the motion of a ball in trajectory flight.

To see an interactive view of two objects falling with the same acceleration, but having different horizontal motions, consider the program below. You can adjust the horizontal speed to the right of each ball (ball A is red and ball B is blue) so that, as it falls, it includes a constant horizontal speed and a vertical acceleration. In particular, note that no matter what horizontal speeds you give, the vertical motion of the two objects is always the same. The two objects are always at the same vertical height at the same time no matter what the horizontal part of the motion is doing.

We can see how all this works quantitatively by using a demonstration in Maple. Note that this demonstration uses animation.

#Start by assuming that vertical and horizontal motion are independent.
#If this is true, then motion along the horizontal axis, which we will
#call x is given by x = x0 + v0x*t + 1/2*ax*t^2.  For simplicity, we
#will start by saying that the velocity along the x axis is constant,
#hence we specify that ax = 0.  We will also start our particle from
#the origin, so x0 = 0.  Therefore, the equation describing x motion is
#given by x = v0x*t, with t being the time.  If the motion along the
#vertical axis is given by z = z0 + v0z*t + 1/2*g*t^2, then we know that
#z0 = 0 (the particle starts at the origin) and g = -9.8 (in units of
#meters/second^2).  For simplicity, let's start with v0z = 0 also so that
#we have the same kind of vertical motion we've been used to seeing in the
#one-dimensional free-fall problems.  Therefore, z = 1/2*g*t^2.  These
#equations can be used to parametrize the trajectory.  Let's animate them 
#to see what they do.  We will include two other particles, one moving only
#along x and one moving only along z to see exactly what our equations of
#motion are doing.  Let's give an initial horizontal speed of 10 meters/sec.
#to our trajectory particle.  Here's our Maple input:
with(plots):
xpos := v0x*t*u;
zpos := 0.5*u*g*t^2;
v0x := 10;
g := -9.8;
animate3d({[xpos, 0, zpos], [-5, 0, zpos], [xpos, 0, 5]},
p=0..5, t=0..5, u=0..1, labels=[x, y, z], frames=15);
#
#Notice that the first square bracket parameterizes the particle of interest,
#the second square bracket parametrizes a particle simply falling with no
#motion in x while the third bracket parametrizes a particle simply moving
#along x with no motion in z.  Amazingly, the first square bracket, which
#we recognize intuitively as the path of an object thrown straight away from
#us, is just the combination of the horizontal and vertical motion!