How long will an iron cannon ball thrown from a boat take to reach the deepest point of the ocean?

Question

Deepest point of the ocean is the Mariana Trench with 10,911 m (35,797 feet) below sea level. Assume the cannon ball is perfectly round and weights 30 pounds = 13.6 kg. Make reasonable assumptions for friction coefficients, floating forces and other parameters.
Just to be clear....I don't know the answer myself but would like to know it :-)

Answer 1

This is an oddly specific question just to be asked out of curiosity.

The mass of the cannon ball is 13.6 kg, thus it has a weight of about 133.4 Newtons (assume g = 9.81 m/s^2).
The force of weight is pointed down...however, since the cannon ball in not in a vacuum and/or not a point mass, it has a non-zero buoyant force acting upward on it.

The buoyancy force acting on the ball is dependent on the weight of the fluid (water in this case) it displaces.
Iron has a density of about 7874 kg/m^3, thus a 13.6 kg sphere would have a volume of about 1.73 E-3 m^3.
If we assume the salt water of the ocean has a uniform density of about 1028 kg/m^3, then the weight of the displaced water exerts a force of about 17.42 Newton’s upward.

Thus, the net force acting on (stationary) cannon ball is about 116 Newtons downward.
According to Newton’s 2nd law,
F = ma
This force will cause the 13.6 kg cannon ball to accelerate at a rate of 8.53 m/s^2.

Assuming there is no "water" resistance / drag, then using the formula,
d = 1/2 at^2,
where the ball stars from rest and experiences a uniform acceleration for a time t as the ball covers d distance,
The cannon ball should take about 50.57 seconds to reach the bottom.

HOWEVER,
Just as there would be air friction if this ball was launched into the air, there is most defiantly "water" friction as this ball accelerated downward....eventually reaching a terminal velocity.
The water will exert a drag force upward on the ball, causing its acceleration to be non-uniform.

The drag force can be calculated as,
F_d = C_d * ((p * v^2) / 2) * A
where F_d is the drag force, C_d is the drag coefficient, p is the density of the medium which the ball is falling in, v is the instantaneous velocity of the ball, and A is the cross sectional surface area of the ball in the direction of travel.
In order to calculate the drag force, one must know the drag coefficient of the cannon ball in water, something which usually must be calculated experimentally.
The ball will accelerate non-uniformly until it reaches it terminal velocity in the water, at which time, acceleration will cease and the ball will continue falling at its terminal velocity.

Unless more information is know (the drag coefficient), the best (guess)/estimate of the how long the ball will take is sometime longer than 50.57 seconds...probably on the order of well over a minute since the ball will not be accelerating nearly as long as it did in our mathematical model calculate earlier.

Answer 2

That only happens if u throw it on earth. Why? Because there is air, which causes resistance (something like friction). Air resistance is an upward force which acts against the gravitational force of the earth acting on the object concerned. The downward force is given by the formula W=mg, which is the mass times gravitational acceleration. As an object falls, its acceleration lessens, until a point where that falling object is in equilibrium. (which we say that the object has achieved terminal speed, i.e. it is falling at a constant velocity). Now, an object with a greater mass would take much longer a time to achieve terminal speed because its downward force is much greater. this means that it falls with a faster average velocity and would achieve a greater terminal speed. p/s pls check ur textbook to see if i'm correct, i might be wrong...

Answer 3

Using the density of iron and assuming it is a perfect sphere, the ball would have a diameter of nearly 15.4 cm. Assuming a salt water density of 1030 kg/m3, the balance of bouyant versus gravitational forces should come out to be nearly 113.7 N in the downward direction. I estimate the time to be 51 seconds.

Answer 4

Not enough info. You would need to know the size of the cannon itself, its rate of fire and type of explosive used to send the ball on its way.

The angle of the shot depends on distance too.

Answer 5

2 minutes

Answer 6

it would depend on ocean currents and wether or not it landed on a whales back