Projectile motion on Earth

PROJECTILE MOTION ON EARTH

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 on the surface of Earth.

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

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)

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 Earth (9.8 m/s²)

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

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

GRAPH

In order to plot the projectile motion, substitute equations (5) and (7) in equation (1) which is the equation of the projectile motion.

Since the curve passes through the origin, it must satisfy the origin or in other words the origin is a trivial solution of the above equation. Thereby substituting (x, y) as (0, 0) in the above equation, we can determine the value of the focal length which is a constant.

Now substitute equation (11) in equation (9) in order to plot the projectile motion.

Input the above equation in a suitable equation or curve plotter and the corresponding result will be obtained as shown in figure.2

Fig .2 Projectile motion on Earth

CONCLUSION

Thus the time of flight, maximum attainable height and distance of an object undergoing parabolic motion on the surface of Earth were determined successfully. The plot also verifies a proper projectile motion with the maximum attained height and distance.