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Chapter 2: Problem 38

Raindrops. If the effects of the air acting on falling raindrops are ignored, then we can treat raindrops as freely falling objects. (a) Rain clouds are typically a few hundred meters above the ground. Estimate the speed with which raindrops would strike the ground if they were freely falling objects. Give your estimate in \(\mathrm{m} / \mathrm{s}, \mathrm{km} / \mathrm{h},\) and \(\mathrm{mi} / \mathrm{h}\) . (b) Estimate (from your own personal observations of rain) the speed with which raindrops actually strike the ground. (c) Based on your answers to parts (a) and (b), is it a good approximation to neglect the effects of the air on falling raindrops? Explain.

Short Answer

Expert verified
Ignoring air resistance is not a good approximation as actual speeds are much slower.

Step by step solution

01

Understand the Problem

We need to determine the estimated speed of freely falling raindrops from clouds. For this, we assume free fall motion without air resistance.
02

Apply Free Fall Formula for Part (a)

For an object in free fall, the final velocity \( v \) is related to the initial velocity \( v_0 \), the height \( h \), and acceleration due to gravity \( g \). The formula is \( v = \sqrt{2gh} \). Assume \(v_0 = 0\), \(g = 9.81 \, \mathrm{m/s^2}\), and \(h = 300 \, \mathrm{m}\) as typical cloud height.
03

Calculate Velocity in m/s

Substitute into the formula \( v = \sqrt{2 \times 9.81 \, \mathrm{m/s^2} \times 300 \, \mathrm{m}} \). Calculate the value.
04

Convert Velocity to km/h

Since 1 m/s = 3.6 km/h, multiply the result from Step 3 by 3.6 to convert the velocity from m/s to km/h.
05

Convert Velocity to mi/h

Since 1 m/s = 2.237 mi/h, multiply the result from Step 3 by 2.237 to convert the velocity from m/s to mi/h.
06

Estimate Actual Velocity for Part (b)

Based on observations, raindrops generally fall at about 8-10 m/s, which is significantly slower due to air resistance.
07

Evaluate Approximation for Part (c)

Compare the speeds from Parts (a) and (b) to decide if neglecting air resistance was a reasonable assumption. The significant difference suggests it is not a good approximation.

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Key Concepts

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

Air Resistance
When objects fall through the air, they are subject to a force termed air resistance, also known as drag. This force opposes the motion of the object. Unlike the constant force of gravity, air resistance increases with velocity. As a result, the faster an object moves, the greater the air resistance it encounters.

In the case of raindrops, air resistance is crucial in reducing their speed as they fall from the clouds to the ground. Without air resistance, raindrops would accelerate due to gravity until they reached high speeds, causing them to hit the ground much faster.

This resistance explains why the speed of falling raindrops is significantly less in practice than calculated in a vacuum. When raindrops fall, they quickly reach a terminal velocity, where the force of gravity is balanced by the air resistance, and they stop accelerating. This terminal velocity for raindrops is much lower than what would be calculated assuming no air resistance.
Velocity Conversion
Velocity, which is the speed of an object in a given direction, can be measured in various units. Commonly used units include meters per second (m/s), kilometers per hour (km/h), and miles per hour (mi/h).

Converting between these units is essential for comprehending speeds in different contexts. To convert meters per second to kilometers per hour, multiply by 3.6, as there are 3.6 kilometers in each meter per second. Conversely, to convert from meters per second to miles per hour, use the conversion factor of 2.237.

Using these conversions, you can easily express velocity in the unit most convenient for your needs. For example, from the exercise, knowing that raindrops fall at around 10 m/s can be interpreted as 36 km/h or approximately 22.37 mi/h.
Acceleration Due to Gravity
Gravity is a universal force that attracts objects towards each other. On Earth, this force imparts an acceleration to falling objects, known as gravitational acceleration, usually denoted as "g". The standard value for gravitational acceleration on Earth is approximately 9.81 meters per second squared \((\mathrm{m/s^2})\).

In free fall, the only force acting on an object is gravity, causing it to accelerate towards Earth at this constant rate. This can be expressed mathematically through the formula:\[ v = \sqrt{2gh} \]

In this formula, "\(v\)" denotes the final velocity reached, "\(g\)" is the acceleration due to gravity, and "\(h\)" is the height from which the object falls. It鈥檚 crucial to use this formula for estimating the speed of objects in free fall, like raindrops in this exercise, assuming no other forces like air resistance are at play.

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Most popular questions from this chapter

The acceleration of a particle is given by \(a_{x}(t)=\) \(-2.00 \mathrm{m} / \mathrm{s}^{2}+\left(3.00 \mathrm{m} / \mathrm{s}^{3}\right) t .\) (a) Find the initial velocity \(v_{\mathrm{ax}}\) such that the particle will have the same \(x\) -coordinate at \(t=4.00 \mathrm{s}\) as it had at \(t=0\) (b) What will be the velocity at \(t=4.00 \mathrm{s} ?\)

The position of the front bumper of a test car under micro-processor control is given by \(x(t)=2.17 \mathrm{m}+\left(4.80 \mathrm{m} / \mathrm{s}^{2}\right) t^{2}-\) \(\left(0.100 \mathrm{m} / \mathrm{s}^{6}\right) t^{6} .\) (a) Find its position and acceleration at the instants when the car has zero velocity. (b) Draw \(x-t, v_{x}-t,\) and \(a_{x}-t\) graphs for the motion of the bumper between \(t=0\) and \(t=2.00 \mathrm{s}\)

The acceleration of a bus is given by \(a_{x}(t)=\alpha t,\) where \(\alpha=1.2 \mathrm{m} / \mathrm{s}^{3} .\) (a) If the bus's velocity at time \(t=1.0 \mathrm{s}\) is 5.0 \(\mathrm{m} / \mathrm{s}\) , what is its velocity at time \(t=2.0 \mathrm{s} ?(\mathrm{b})\) If the bus's position at time \(t=1.0 \mathrm{s}\) is \(6.0 \mathrm{m},\) what is its position at time \(t=2.0 \mathrm{s} ?\) (c) Sketch \(a_{x}-t, v_{x}-t,\) and \(x-\) graphs for the motion.

A subway train starts from rest at a station and accelerates at a rate of 1.60 \(\mathrm{m} / \mathrm{s}^{2}\) for 14.0 \(\mathrm{s}\) . It runs at constant speed for 70.0 \(\mathrm{s}\) and slows down at a rate of 3.50 \(\mathrm{m} / \mathrm{s}^{2}\) until it stops at the next station. Find the total distance covered.

A student throws a water balloon vertically downward from the top of a building. The balloon leaves the thrower's hand with a speed of 6.00 \(\mathrm{m} / \mathrm{s}\) . Air resistance may be ignored, so the water balloon is in free fall after it leaves the thrower's hand. (a) What is its speed after falling for 2.00 s? (b) How far does it fall in 2.00 \(\mathrm{s?}\) (c) What is the magnitude of its velocity after falling 10.0 \(\mathrm{m} ?\) (d) Sketch \(a_{y}-t, v_{y}-t,\) and \(y-t\) graphs for the motion.
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