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.

Projectile motion on Black Hole

PROJECTILE MOTION ON BLACK HOLE

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 determine time of flight, maximum attainable height and maximum attainable distance of an object undergoing projectile motion.

ASSUMPTIONS

1. Air, wind and other frictional resistance are neglected

2. Effect of rotation of Black hole 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 even after crossing the event horizon

7. The motion is always parabolic without any effects from space-time distortion

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 Black hole. 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 black hole is a collapsed star which has infinite density at the singularity. The acceleration due to gravity value depends on the type of black hole being considered. But in general the tidal forces are so strong that it is appropriate to assume infinite gravitational acceleration.

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)

A projectile motion is represented in figure.1 with all the coordinates

Fig .1 A projectile motion

Now, the other aspects of the parabolic motion such as total time taken, maximum height and maximum distance attained will be discussed

The time of flight ‘T’ is given by,

where,

T – Time of flight or total time taken (s)

u – Initial velocity of projectile (m/s)

θ – Angle of projectile (degree)

g – Acceleration due to gravity on Black hole (ꝏ m/s²)

The value ‘0’ indicates that time does not exist inside a black hole. This is because of extreme gravitational forces inside a black hole.

The maximum height attained (H) by the object is given by,

The value ‘0’ indicates that the projectile motion will never start since time is frozen on a black hole.

The maximum distance attained (d) by the object is given by,

The value ‘0’ again indicates that there will be no projectile motion.

Equations (3), (5) and (7) indicate that projectile motion does not occur due to extreme gravitational forces inside a black hole.

CONCLUSION

Thus it can be observed from the calculations that the projectile motion does not occur inside a black hole.