Question

modern applications of keplers calculations are used today by nasa scientists nasa has sent many satellites, landers, and rovers to mars beginning with the first landing of viking 1 on the surface of mars in 1976. the model below represents the flight path for the curiosity rover after being launched from earth on november 26, 2011. the distance between tick marks on the flight path is 20 earth days. flight path of curiosity rover use the model to calculate the approximate travel time between earth and mars in earth days. construct an explanation that describes why this flight path was selected rather than a straight - line path to mars on the launch date. a. approximate travel time: earth days b. flight path explanation:

Explanation:

Step1: Count the number of tick - marks

Count the number of intervals (tick - marks) on the flight path. There are 10 intervals.

Step2: Calculate the travel time

Since each interval represents 20 Earth days, the travel time \(t\) is given by the formula \(t=\text{Number of intervals}\times20\). So \(t = 10\times20\).

Answer:

a. \(200\) Earth days
b. A straight - line path would require a large amount of fuel to overcome the gravitational pull of the Sun and to change the velocity of the rover significantly. The chosen flight path (a Hohmann transfer orbit - which is an elliptical path) uses the gravitational forces of the Sun and the orbital motion of Earth and Mars more efficiently. It allows the rover to use less fuel as it “hitches a ride” on the orbital dynamics of the planets around the Sun. This is in line with Kepler's laws of planetary motion, which describe the elliptical orbits of planets. The rover can take advantage of the conservation of angular momentum and the gravitational potential energy changes in this elliptical path.