# I Speed of falling object underwater

#### sadidrahman20

Is there any formula for underwater falling sphere which can measure the velocity of that sphere at any time before it reaches its terminal velocity? A sample question is: I dropped a sphere into water with 0m/s initial velocity. what will be the velocity of that sphere after 3s? {the radius and density of that sphere is .005m & 2500kgm-3, the viscosity coefficient of water is .00016Nsm-2.}

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#### A.T.

Science Advisor
Is there any formula for underwater falling sphere which can measure the velocity of that sphere at any time before it reaches its terminal velocity?
It's like falling in air, just that more gravity is balanced by buoyancy. This is for linear drag:

#### jbriggs444

Science Advisor
Homework Helper
You might be thinking that when a falling object falls, the surrounding fluid also rises. It follows that an accelerating fall involves the downward acceleration of the falling object and also the upward acceleration of the surrounding fluid.

One would need a rule of thumb to decide how much of the surrounding fluid is affected in this manner to come up with a reasonable split for the potential energy being released going into body and fluid kinetic energy respectively. Experiment rather than theory should be the guide in this case. [Probably child's play for computational fluid dynamics].

#### jrmichler

Science Advisor
If the water is filling a cylinder with inside diameter exactly the same as the outside diameter of the sphere, the calculation is real simple: the sphere plugs the tube and stops.

If the water is in a container much larger than the sphere (think swimming pool), then you can use numerical methods. A free body diagram (FBD) of the sphere has gravity force, buoyancy force, acceleration, and fluid drag. The fluid drag is a function of velocity. To calculate drag, search Reynolds Number and Drag Coefficient Sphere. Calculate all of the forces, then iterate with small time steps. The drag force needs to be recalculated after each time step. The time steps need to be small enough that the velocity does not change significantly between steps.

Hint: The problem is easier if you start with the sphere fully submerged at zero velocity, then let go.

It's a fun little problem. Enjoy.

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