©Fernando Caracena 5 August 2012
A portrait of Galileo Galilei by Justus Sustermans (1597–1681). From the Commons.
Galileo investigated motion experimentally and came up with quantitative relationships (formulas) for uniform and accelerated motion, which is a branch of physics called kinematics. A lot of his thinking was a combination of geometry, algebra and numerical analysis. His studies of motion resulted in several discoveries:
- Each motion along a set of three orthogonal axes in three dimensional space can be described independently as a one-dimensional motion.
- There exist unaccelerated (inertial) frames of reference in which the laws of physics are the same (Galilean Relativity).
- The laws of accelerated motion.
In grokking Galileo's laws of motion, the following discussion uses arguments from geometry combined with simple algebra. It follows some of the intuitive ideas that come from plotting functions on graph paper. Basic to the arguments presented here are the ideas associated with plotting linear functions. Plotting functions gives you the experience that you need to grasp the intuitive concepts presented here. Lines plotted on Cartesian coordinates in two dimensions satisfy the rules of ordinary Euclidean geometry. This geometry gives you a means of visualizing equations; and visualization brings you to the brink of understanding.
First of all, let us review how to calculate the area of two, simple, geometric figures of geometry, the rectangle and the right triangle.
The area of a rectangle(A), which has a base, b, and height h is given by
A= b * h . (1)
This equation is almost self evident. Just look at the squares on qaudrille paper, select any rectangle that runs along the rullings, then count squares. Voila! The results will verify (1) every time. [Note the terminology used here. (1) means equation 1.] There is a way of proving (1) formally by induction, but we will not go into the details here, since we are grokking. Everyone who wants to do practical work, such as tiling a bathroom, should know the relationship expressed in (1). Otherwise, tiles may run out in the middle of a project, or there may be surplus tiles.
A rectangle splits into two right triangles when a diagonal line connects opposite corners. In the figure, the top rectangle is split into two right triangles of equal area, one of which is painted a solid black, the other gray.
Ignore the elongated, lower rectangle for the moment. Since the black and gray triangles' areas add up to those of the top rectangle and they are equal in area, the area of the black triangle is half that of the top rectangle:
Atri =(1/2)*b*h . (2)
Now, that is almost all the mathematical background that we need to grok Galileo's physics.
Consider what the lower, gray rectangle corresponds to. It represents the distance traveled at a constant velocity (Vo) after a time (t) as the area swept out in that rectangle by the increase in time,
D = Vo*t. (3)
The increasing area of v vs. time figure is the distance traveled. The area of the black triangle in the figure is just the distance traveled under the variable speed, v(t), which starts from rest and increases linearly with time in the figure:
D=(1/2)*v(t)*t, (4)
where
v(t)=a*t (5)
and a is a constant (acceleration, also the slope of the line gained by plotting (5)).
Substituting for v(t) from (5) in (4) results in an equation for the distance traveled from rest under constant acceleration:
D=(1/2)*a*t2. (6)
Note that (6) can be modified to describe the total distance traveled from some initial time, if you begin with an initial velocity Vo, instead of starting from rest, by adding the area of the lower rectangle (Vo*t) to the right hand side (RHS) of (6):
D'=Vo*t + (1/2)*a*t2. (7)
Note that in this case, the RHS of (7) is the area between the line, V(t), and the horizontal bottom of the figure.
Next go to Grokking Galileo's Physics II.