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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. We've already considered free-fall in one dimension, along the vertical axis. Now, the next step would be to ask what happens when we combine motion along the horizontal AND vertical axis. Nature is particularly kind to us in that there is really no complication at all, at least not mathematically. When we throw an object into the air, our eyes trace its path or trajectory. The characteristic shape of the trajectory is called parabolic. We can now explore why this is, by noting from examples shown in class that motion along the vertical and horizontal axes are independent, 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. We can see how this works by demonstration in Maple. Note that this demonstration uses animation. It can only be done if you have access to a Windows machine running Maple V, version 2. If you can't get access to one, don't worry, we will do this demo in class.
#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!