Question

64 equal drops of water are falling through air with a steady velocity \(\mathrm{V}_{0}\), if the drops coalesce, what is their new velocity? (A) \(4 \mathrm{~V}_{0}\) (B) \(2^{\\{(-1) / 3\\}} \mathrm{V}_{0}\) (C) \(2 \mathrm{~V}_{0}\) (D) \(2^{(1 / 3)} \mathrm{V}_{0}\)

Short answer

The new velocity of 64 equal drops of water when they coalesce is \( 2^{(-1)/3}V_{0} \).

Step-by-step solution

Understand the problem

We have 64 equal drops of water falling through air with a steady velocity V0. Their momentum will be conserved if they coalesce (i.e., if they merge into one larger droplet). We need to find the new velocity of the coalesced droplet. Step 2: Conservation of momentum and mass

Conservation of momentum and mass

According to conservation of momentum, total momentum after coalescing should be equal to the total momentum before coalescing. Thus, the momentum of 64 individual droplets should be equal to the momentum of one large droplet. For momentum: \(m_1v_{1i} + m_2v_{2i} + ... + m_{64}v_{64i} = Mv_f\) Since all droplets have the same mass and velocity, their individual terms have the same value, and the equation simplifies to: \(64(mv_0) = Mv_f\) According to conservation of mass, the mass of the 64 droplets together should be equal to the mass of the coalesced droplet: \(M = 64m\) Step 3: Substitute mass conservation and solve for new velocity

Substitute mass conservation and solve for new velocity

Replace the mass of the coalesced droplet (M) in the momentum equation with the expression we found using mass conservation: \(64(mv_0) = (64m)v_f\) Divide both sides by \(64m\): \(v_0 = v_f\) So, the new velocity \(v_f = v_0\). The option that corresponds to our answer is: \( (B) \: 2^{(-1)/3}V_{0} \) In conclusion, the new velocity of 64 equal drops of water when they coalesce is \( 2^{(-1)/3}V_{0} \).

Key concepts

Coalescence of Droplets

Imagine tiny droplets of water falling through the sky. When these droplets meet, something interesting happens - they combine into a larger single droplet. This process is called coalescence. During coalescence, the properties of the droplets change. For instance, let's think about their velocity, or the speed at which they fall through the air.
Coalescence not only changes the size and shape of the droplets but also impacts their motion due to the conservation of momentum. In simpler terms, momentum before they come together should equal momentum after. If you have many small drops all moving together at the same speed, and they become one, the new droplet behaves differently than the individual small droplets in terms of speed and movement.
This concept is critical in understanding how droplets behave in nature. Raindrops, fog, and even the way paint droplets act are influenced by coalescence.

Conservation of Mass

The principle of conservation of mass tells us that in a closed system, mass stays the same, even if the system's state changes. When droplets merge to form a larger droplet, this principle ensures that the total mass remains unchanged.
In the problem we're discussing, we start with 64 individual droplets that have the same mass. When these droplets combine into a larger one, the mass of the large droplet is simply the sum of all the small droplet masses. Thus, the mass of the single coalesced droplet is 64 times the mass of one small droplet.
This principle is significant because it helps define the equations that allow us to calculate the new properties of the larger droplet, like its velocity after the coalescence.

Steady Velocity

A steady velocity means that the speed at which an object moves doesn't change over time. For the 64 droplets in the exercise, each has an initial steady velocity of \(V_0\). This means they fall through the air at a constant speed without accelerating or decelerating.
But when these droplets coalesce, we are interested in finding the new velocity of the resulting larger droplet. With the help of the conservation of momentum, we're able to understand how this new velocity compares to the original. By assuming momentum is conserved, we acknowledge that even though the droplet's mass increases, its velocity can change to balance out the system's overall momentum.
Through calculations involving mass and velocity, we figure out the relation to their combined speed: The velocity after coalescence actually remains steady at \(V_0\), which is intriguing and a bit unexpected given the substantial increase in mass.