Comparison of projectile motions on cosmic bodies

COMPARISON OF PROJECTILE MOTIONS ON COSMIC BODIES

INTRODUCTION

Projectile motion is a form of motion that follows or traces out a parabolic path. The predominant reason for the origin of projectile motion is acceleration due to gravity. When an object is simply given a horizontal initial velocity, gravity which is always acting downward will exert a vertical pull on the object. The resultant of horizontal and vertical components is a parabolic motion. The magnitude of horizontal and vertical components may or may not be equal since it depends on the angle of trajectory. However both horizontal and vertical components are totally independent of each other. In this post, we intend to compare the trajectories of a projectile on cosmic bodies such as Moon, Earth, Jupiter and Sun.

ASSUMPTIONS

1. Air, wind and other frictional resistance are neglected

2. Effect of rotation of earth is negligible

3. Temperature effects do not impede the motion

4. The ground surface is perfectly horizontal

5. The projectile moves along a two dimensional path

6. The projectile is indestructible

CALCULATION

Consider an object of mass ‘M’ kg, moving with an initial velocity ‘u’ at an angle ‘θ’ with respect to the horizontal. Let ‘g’ be the acceleration due to gravity on Earth. It is important to note that projectile motion is independent of the mass of the object in a vacuum. However in air or other media, the drag coefficient being different for various object shapes and sizes, it is no longer independent of mass.

The equation of projectile motion in this case is given by,

where,

h – Horizontal distance at which the projectile attains maximum height (m)

k – Maximum height attained by the projectile (m)

a – Focal length of the parabola (m)

As discussed in the previous posts, the time taken, maximum height and distance attained are different for each planet or star. The equations of motion for all the cosmic bodies with the constants are given in table.1

GRAPH

The equations in table.1 are plotted and the graph is as follows:

This graph allows us to study the effect of acceleration due to gravity on the profile of the trajectory undergoing projectile motion. The trajectory travels maximum distance and achieves maximum height on Moon due to low acceleration due to gravity while it is the opposite on Sun.

CONCLUSION

Thus the trajectories of projectile motion on various cosmic bodies were successfully plotted and studied.