Engineering Expert Witness Blog

      Last time we introduced Newton’s equation to calculate the sun’s gravitational force acting upon Earth, and today we’ll begin solving for the last remaining unsolved variable within that equation, v, Earth’s orbital velocity.

      Here again is Newton’s equation,

F_g = [m × v²] ÷ r

      For a refresher on how we solved for m, Earth’s mass, and r, the distance between Earth and the sun, follow these links to past blogs in this series, What is Earth’s Mass and Calculating the Distance to the Sun.

      Velocity, or speed, as it’s most commonly referred to, is based on both time and distance.   To bear this out we’ll use an object and situation familiar to all of us, traveling in a car.   The car’s velocity is a factor of both the distance traveled and the time it takes to get there.  A car traveling at a velocity of 30 miles per hour will cover a distance of 30 miles in one hour’s time.   This relationship is borne out by the formula,

v_Car = distance traveled ÷  travel time

v_Car = 30 miles ÷ 1 hour = 30 miles per hour

      Similarly, v is the distance Earth travels during its orbital journey around the sun within a specified period of time.   It had been observed since ancient times that it takes Earth one year to complete one orbit, so all that remained to be done was calculate the distance Earth traveled during that time.   Vital to calculations was the fact that Earth’s orbit is a circle, which allows geometry to be employed and calculations to be thereby simplified.   Refer to Figure l.

Figure 1

      From geometry we know that the circumference of a circle, C, is calculated by,

C = 2× π × r

where π is a constant, the well known mathematical term pi, which is equal to 3.1416, and r is the radius of Earth’s circular orbit, determined, courtesy of the work of Johannes Kepler and Edmund Halley, to be approximately 93,000,000 miles.   Stated in metric units, the unit of measurement most often employed in science, that comes to 149,000,000,000 meters.

      Inserting these numerical values for π and r into the circumference formula, scientists calculated the distance Earth travels in one orbit around the sun to be,

C = 2 ×  π × 149,000,000,000 meters = 9.36 x 10¹¹ meters

      Next time we’ll introduce the time element into our equations and solve for v, and from there we’ll go on and finally solve for F_g, the sun’s gravitational force acting upon Earth.

____________________________________