Question 1: Eight equal drops of oil are falling through air with steady velocity of 0.1 m/s. The drops recombine to form a single drop, what should be the new terminal velocity?

Solution

Given

No of small drops = 8

Velocity of each individual drop = 0.1 m/s

No of large (combined) drops = 1

Find

Terminal velocity of the combined drop

Formulae

(1) Terminal velocity =

(2) Volume of the sphere,

Assumptions: We make a number of assumptions. Let,

Mass of the small drop = m Volume of the small drop = V Radius of the small drop = r

Mass of the combined drop = M Volume of the combined drop = V_{c} Radius of the combined drop = R Terminal velocity of the combined drop = V_{cT}

Comprehension:
The eight drops are of the same fluid and having the same density. Therefore, when they combine together, the mass and volume of the larger drop will be,
M = 8m and V_{c} = 8V. However, the radius of the combined drop will NOT be 8r. This is because the volume and radius has not a linear relation.

Solution

For the solution, we need to find the radius R of the combined drop and then put it in the formula 1 to find the terminal velocity of the combined drop. For the calculation of R, we use formula 2.

Put values in formula 2 to find the volume V of the small drop,
Similarly, volume of the combined drop,
Now put the value in V_{c} = 8V as calculated in comprehension.

Take cube root of the eqution

Put this value in Formula (1) for the terminal velocity of the combined larger drop.

But M = 8m, therefore,

On the RHS, the value in bracket is the terminal velocity of the small droplets. So put it,

Where v = 0.1 m/s is the velocity of the small drop given in the text.

Follow up

Since the density of the individual droplets and the big drop is same, we also can apply the equation . Use this equation to solv the problem.

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