Reference no: EM131005798

Activity Why Do We Survive Rainy Days,

"Puddles of water are stalling to form everywhere, but I slither carefully around them on the way to my car. Meanwhile, thousands of raindrops are splatting all over my umbrella. Vet somehow, umbrella is keeping me safe from these hundreds of drops falling from thousands of feet. This isn't particularly surprising at fast. But then I start thinking about the journey that just one of these drops makes. A typical raindrop falls from an average height of 13,000 feet. Along its fall it combines with other droplets, much like snowballs do in cartoons as they roll down the slide of a mountain. This process, known as coalescence, increases the droplet's size and mass. As it falls, the droplet's speed also increases. So, if both the droplet's mass and. speed are increasing as it falls, how male it doesn't crash right through my umbrella? And how do you figure that I'm still alive after umbrella is bombarded by thousands of these raindrops? There must be a way that derivatives eau help us answer these questions."

1. Think about a raindrop as a sphere. The droplet fuses with other droplet as it falls, increasing its mass. As its mass increases, the droplet becomes more likely to fuse with other droplets. Let's denote the mass after some elapsed time, t, by m(t). There which a droplet's mass increases depends on how massive it currently is. Express the previous sentence in mathematical notation and explain the notation you choose.

2. Is the rate at which a droplet's mass increases positive or negative? What are the implications of your response when thinking about the function m(t)?

Momentum depends upon the mass of the abject and the velocity at which the object is moving, For example, aim, truck traveling at a high speed has much momentum- that is, it would take much force to bring this truck to a stop. The same truck traveling at a low speed would be easier to stop and therefore has less momentum. Momentum is defined as the product of the mass and velocity of an object.

3. Again, we denote the mass of our rain droplet by m(t). We can denote its velocity as v(t). 3. Therefore, express the droplet's momentum and denote it using p(t). Describe possible units that could he used to describe the droplet's momentum.

4. Calculate p'(t)and write a description of what information this provides.

5. It can be shown using the result in #4 that v(t) = 32 (1 - e^-2.3t) / 2.3 feet per second. Graph this function and use the graph to determine lim_t→∞ v(t). What information does this provide about the situation involving the rain droplet?

6. Convert the magnums peed of the rain droplet to miles per hour.

"...as a droplet falls and, size increase, it's surface area also increases (recall the snowball analogy).'

"..this (larger) surface area experiences more air drag slowing the acceleration of the droplet. Eventually it reaches an acceleration of zero meaning that its velocity stops increasing. This is what we call terminal velocity and is the approximately 13.92 ft/s (or 9.5 miter)'

'But what about the whole coalescence thing? Wasn't the droplet mass with seemingly nothing to stop it? Here again air resistance saves the day. The resistance the droplet experience breaks is up into smaller parts multiple times during its fall. By the time it hits my umbrella a typical droplet weighs only about 0.0007 pound. With such a tiny mass, each raindrops momentum is just enough to "splat" on my umbrella, but nothing more."