Newton’s Cannonball

Issac Newton was on a quest to explain two aspects of the moon orbiting around the earth. Firstly, the moon should crash to the ground if all objects accelerate towards the earth’s centre. Secondly, what makes the moon move in circular orbits rather than a straight line even though, according to Galileo Galilei, objects tend to move in the same direction and speed until an external force acts upon them. 

Hence, to understand the orbital motion of objects, Newton proposed a thought experiment in 1968 called Newton’s cannonball experiment, which was mentioned in Principia Mathematica.

Newton’s cannonball

Newton’s cannonball experiment was based on a ‘thought experiment’ conceived to understand the orbital motion of objects. In the experiment, a sphere having a uniform density is used as a model of earth. A cannonball is imagined as being fired from a high mountain cannon. Each time, the muzzle velocity of the cannonball is increased till they orbit the earth. The trajectories of the cannonballs describe the orbital tunnels.

If the air resistance is negligible, Six outcomes depend on the cannonball’s velocity. They are:

• The cannonball hits the ground

Imagine a cannonball is fired parallel to the surface of the earth from a cannon at a high mountain. It will cover some distance on the horizontal axis until it hits the ground. This is the displacement of the cannonball, which depends on the initial horizontal velocity with which the cannonball was fired and the time taken by the cannonball to hit the ground from the given height. 

• The cannonball exhibits orbital motion

On increasing the muzzle velocity of the cannonball, at a certain initial horizontal velocity, the cannonball will not fall back on the ground. Still, it will enter the orbit to circle the earth parallel to its surface due to the earth’s gravitational pull.

• The cannonball exhibits an initial elliptical path

On increasing the initial velocity of the cannonball, the length of its elliptical path increases such that the cannonball starts orbiting the earth. This makes the upper axis of the elliptical path coincide with the lower axis of the earth’s centre.

• The cannonball exhibits a circular orbit

On further increasing the velocity of the cannonball, at a certain initial horizontal velocity, the upper and lower axis of the ellipse will coincide with the centre of the earth. Hence, the cannonball moving in an elliptical path will move in circular orbits. 

Newton’s cannonball formula determines the initial velocity required to make the cannonball enter a circular orbit, 

Vc = √gR

Vc is the initial velocity in ms-1 required by the cannonball to enter a circular orbit. 

g = 9.8 ms-2 is the acceleration due to gravity.

R = 6.371 x 106 m is the radius of the earth.

Newton’s cannonball works similar to the moon orbiting in a circular path around the earth. Due to inertia, an object continues to move in a constant direction and speed until it is accelerated. The gravitational pull of the earth provides acceleration in the case of the moon.

• The cannonball exhibits an elliptical orbit

On further increasing the cannonball’s initial velocity, the ellipse’s upper axis approaches the centre of the earth while its lower axis moves further away. This results in the cannonball exhibiting an elliptical orbit.

The initial velocity for a cannonball to exhibit an elliptical path should be more than that for a circular path but less than the escape velocity of the earth.

√gR < Vc < √2gR

 Where gR < Vc means that the cannonball’s velocity is more than that required for a circular path.

 Vc < √2gR means that the cannonball’s velocity is less than the escape velocity of the earth.

• The cannonball flies off into space

If the velocity of the cannonball exceeds the escape velocity of the earth, it flies off into space, which is expressed as, 

Conclusion

The cannonball is similar to the moon in Newton’s cannonball experiment, which explains the circular orbital motion of the moon around the earth, also influencing Newton’s universal law of gravitation. The rate of fall of the moon matches the curvature of the earth, making the moon orbit around the earth in a circular path, just like Newton’s cannonball does. The moon’s orbital motion results from its tangential velocity accelerated by the gravitational pull of the earth.