Calculus: The Language of Change

Exercise Set 10.2
Galileo's Law of Gravity

by John Robeson

Exercise 1) Galileo's Law and (((d^2) s)/(dt^2))

Write Galileo's Law, "The rate of increase in the speed of a falling body is constant" in terms of the derivatives of the distance function s[t].  What derivative gives the speed?  What derivative gives the rate at which the speed increases?

Solution

The rate of increase in the speed is the acceleration which is (((d^2) s)/(dt^2)).  For that to be constant means we can write the law as

		(((d^2) s)/(dt^2)) = g where g is a constant.
		
In other words if we measure the speed at any instant and then again one second later, the reading on our "speedometer" will have increased by an amount equal to g.

The first derivative (ds/dt) gives us the falling speed.

The second derivative(((d^2) s)/(dt^2)) gives us the rate at which the falling speed increases.

Exercise 2) Galileo's Law and the graph of  (ds/dt)

Galileo's Law is easiest to confirm with the data of the Gravity program by looking at the graph of (ds/dt) (because error measurements are magnified each time we take differences of our data).  What feature of the graph of velocity is equivalent to Galileo's Law?

Solution

Evaluate the cell below for a graph of velocity vs. time.

velocitydata={{0.25`,2.44600000000000017`},{0.75`, 7.35599999999999987`},{1.25`, 12.258`},{1.75`, 17.1400000000000041`},{2.25`, 22.0599999999999951`},{2.75`, 26.9400000000000083`},{3.25`, 31.840000000000006`},{3.75`, 36.760000000000006`},{4.25`, 41.6599999999999948`},{4.75`, 46.5399999999999902`},{5.25`, 51.399999999999979`},{5.75`, 56.4000000000000412`},{6.25`, 61.1999999999999921`},{6.75`, 66.1999999999999921`},{7.25`, 71.0000000000000586`},{7.75`, 76.0000000000000053`},{8.25`, 80.7999999999999651`},{8.75`, 85.7999999999999651`},{9.25`, 90.6000000000000227`},{9.75`, 95.6000000000000227`}};
ListPlot[velocitydata, PlotJoined -> True, 
        AspectRatio -> 1, PlotRange -> {0, 100}, 
        AxesLabel -> {time, velocity}];

This graph is a staight line which means that the slope is constant which in turn means that the accelaeration is constant.  So a linear graph of velocity is equivalent to Galileo's Law.