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A Variable-Mass Raindrop. In a rocket-propulsion problem the mass is variable. Another such problem is a raindrop falling through a cloud of small water droplets. Some of these small droplets adhere to the raindrop, thereby increasing its mass as it falls. The force on the raindrop is $$ F_{\mathrm{ext}}=\frac{d p}{d t}=m \frac{d v}{d t}+v \frac{d m}{d t} $$ Suppose the mass of the raindrop depends on the distance \(x\) that it has fallen. Then \(m=k x,\) where \(k\) is a constant, and \(d m / d t=k u .\) This gives, since \(F_{\text {cut }}=m g_{*}\) $$ m g=m \frac{d v}{d t}+v(k v) $$ Or, dividing by \(k_{*}\) $$ x_{B}=x \frac{d v}{d t}+v^{2} $$ This is a differential equation that has a solution of the form \(v=a t\) where \(a\) is the acceleration and is constant. Take the initial velocity of the raindrop to be zero. (a) Using the proposed solution for \(v\). find the acccleration \(a\). (b) Find the distance the raindrop has fallen in \(t=3.00 \mathrm{~s} .(\mathrm{c})\) Given that \(k=2.00 \mathrm{~g} / \mathrm{m},\) find the mass of the raindrop at \(t=3.00 \mathrm{~s}\). (For many more intriguing aspects of this problem, see K. S. Krane, American Jourmal of Physics, Vol. 49(1981) . pp. \(113-117 .\) )

Step by step solution

01

Solve for acceleration 'a'

Given that \(v = a t\), we differentiate this with respect to time to get \( \frac{d v}{d t} = a\). Substituting this into the equation \(x g = x \frac{d v}{d t} + v^2\), we get: \[ g = a + (a t)^2 \] Given that the initial velocity is zero, at \(t = 0\), we have \(g = a\). Thus, the acceleration \(a = g = 9.8 m/s^2\).

02

Determine the distance fallen at \(t=3.00s\)

Looking at the expression \(v = at\), the distance \(x\) can be found by integrating the equation over time: \[ x = \int_{0}^{t} v dt = \int_{0}^{t} a t dt = \frac{1}{2} a t^2 \] Substituting for \(t = 3.00 s\) and \(a = 9.8 m/s^2\), we find that \(x = 44.1 m\).

03

Find the mass of the raindrop

Given that \(m = kx\), where \(k = 2.00 g/m\), the mass of the raindrop at \(t=3.00s\) can be found by substituting \(x = 44.1m\) into the equation. That gives the mass as \(m = 88.2g\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differential Equations

Differential equations are equations that involve an unknown function and its derivatives. In this particular problem, we see how differential equations are used to describe systems with variable mass. Here, the motion of a raindrop, which gains mass as it falls through a cloud, is modeled using differential equations.
This process involves relating forces and motions (like acceleration and velocity) with the change in mass. Such equations help us understand more complex scenarios in physics, like those involving rocket propulsion or other systems where mass changes over time.
The specific differential equation at hand is \[ x_{B}=x \frac{d v}{d t}+v^{2} \] which ties together different parameters: distance fallen, acceleration, and velocity. By solving these equations, we can determine how the raindrop's velocity and position evolve as it moves.

Rocket Propulsion

Rocket propulsion is a fascinating area that shares similarities with the problem of a raindrop gaining mass. In rocket propulsion, as fuel is expelled, the rocket's mass decreases, yet it can achieve acceleration due to the conservation of momentum.
In the case of variable mass systems like rockets, understanding how mass changes over time is crucial. The principle used is sometimes referred to as Tsiolkovsky's rocket equation, which relates a rocket's change in velocity to its mass and the velocity of the expelled fuel.
Both rocket propulsion and the falling raindrop problem explore the physics of systems where mass is not constant, requiring us to consider how momentum and force equations adapt when mass changes. These principles allow us to solve real-world problems where mass variation plays a significant role.

Motion with Variable Mass

Motion with variable mass is a captivating concept in physics. In our exercise, the raindrop experiences an increase in mass as additional moisture adheres to it during its fall. This phenomenon requires us to approach problem-solving with a different set of tools than we use for constant-mass systems.
When mass is variable, the momentum of the system has to consider both mass and velocity changes, leading to more intricate equations. This understanding allows for precise calculations regarding acceleration, velocity, and displacement over time.

• The fact that the mass depends on the distance fallen introduces a relation, \( m = k x \), which bridges the change in mass to the position.
• This connection means that as the raindrop falls, its mass steadily increases at a rate proportional to the distance it travels.

By accurately incorporating these changes, predictions about motion become more reliable, providing insights into both natural and engineered systems.

Classical Mechanics

Classical mechanics provides the framework for understanding motion and forces in simple and more complex systems, like the raindrop example. This branch of physics uses the principles laid out by Newton to explain the behavior of objects and has been a cornerstone for further advancements.
One of the key insights in classical mechanics is how force relates to an object's motion — specifically through Newton's second law, \( F = m a \). However, when dealing with variable mass systems, such as a growing raindrop, the typical laws slightly morph to incorporate mass change.
Through problems like the falling raindrop, classical mechanics showcases its adaptability. By integrating additional factors, like mass variation, we extend Newton's principles to cover scenarios beyond the constant-mass systems originally envisioned. This not only underscores the depth and flexibility of classical mechanics but also illustrates its enduring relevance to modern physics applications.