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Chapter 2: Problem 98

A Supersonic Waterfall Geologists have learned of periods in the past when the Strait of Gibraltar closed off, and the Mediterranean Sea dried out and become a desert. Later, when the strait reopened, a massive saltwater waterfall was created. According to geologists, the water in this waterfall was supersonic; that is, it fell with speeds in excess of the speed of sound. Ignoring air resistance, what is the minimum height necessary to create a supersonic waterfall? (The speed of sound may be taken to be \(340 \mathrm{m} / \mathrm{s} .\) )

Short Answer

Expert verified
The minimum height is approximately 5908.16 meters.

Step by step solution

01

Understand the scenario

The problem requires us to calculate the height from which water must fall to reach or exceed the speed of sound, ignoring air resistance. The speed of sound is given as \(340 \mathrm{m/s}\).
02

Identify the relevant physics principle

We use the equation for gravitational acceleration to determine the speed of water as it falls freely under gravity. The potential energy of water at height \(h\) is converted to kinetic energy as it falls, leading to the equation: \(v = \sqrt{2gh}\) where \(v\) is the velocity, \(g\) is the acceleration due to gravity (approximately \(9.8 \mathrm{m/s^2}\)), and \(h\) is the height.
03

Set up the equation for the height

We want to find the height \(h\) where the free fall speed \(v\) equals the speed of sound, \(340 \mathrm{m/s}\). Using the formula from Step 2: \(340 = \sqrt{2 \times 9.8 \times h}\).
04

Solve for height \(h\)

Square both sides of the equation from Step 3 to eliminate the square root: \((340)^2 = 2 \times 9.8 \times h\). Simplify to get: \(115600 = 19.6h\). Divide by \(19.6\) to isolate \(h\): \(h = \frac{115600}{19.6}\).
05

Calculate the numerical value

Perform the division to find \(h\): \(h \approx 5908.16 \mathrm{m}\). Thus, the minimum height necessary to create a supersonic waterfall is approximately \(5908.16 \mathrm{m}\).

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Key Concepts

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

Free Fall
Free fall is a fascinating concept in physics that describes the motion of objects falling under the influence of gravity alone. In a free fall situation, objects are only affected by gravity without any interference from air resistance or other forces. Imagine dropping a ball from a cliff; as it falls, it speeds up continuously because of gravity.

In the special case of the supersonic waterfall, the water falls freely, converting its gravitational potential energy into kinetic energy. This transformation allows the water's speed to increase as it keeps falling. The lack of air resistance in our scenario simplifies calculations since we don't have to account for any slowing effects the air might have. This means that the speed of the water only depends on how far it falls, which leads us to the concept of potential energy being fully converted into kinetic energy.

To put it simply, free fall allows us to predict the speed of an object based entirely on the height from which it was dropped. If we know the height, we can calculate the speed using the formula:
  • Potential Energy = Kinetic Energy
  • Gravitational force is the only force acting
  • Gravity = 9.8 m/s虏 on Earth
Supersonic Speed
Supersonic speed means traveling faster than the speed of sound in air, which is approximately 340 m/s under normal conditions. When something moves at supersonic speed, it creates interesting phenomena like shock waves鈥攖hink of the loud boom produced when a jet breaks the sound barrier.

In the context of the waterfall exercise, reaching supersonic speeds implies the water must fall from a significant height. The calculated speed of 340 m/s is the threshold where the water鈥檚 velocity transitions from subsonic (slower than sound) to supersonic (faster than sound). This phenomenon is remarkable as it showcases how energy conversion鈥攖hrough gravity鈥攃an propel the water to such high speeds.

Although achieving such a speed in a waterfall might seem imaginary, understanding it helps reinforce concepts like velocity, energy transformation, and the effects of gravity.
  • Supersonic speed > 340 m/s
  • Requires significant height in free fall
  • Creates shock waves due to speed
Acceleration due to Gravity
Acceleration due to gravity is a constant force that acts on all objects near the Earth鈥檚 surface. It is denoted by the symbol "g" and has an average value of 9.8 m/s虏. This acceleration acts in the downward direction, meaning it pulls everything towards the center of the Earth.

In our exercise, the acceleration due to gravity is crucial because it determines how fast the water can actually fall. When calculating the potential for achieving supersonic speeds in a free fall, it's this acceleration that allows for the conversion of potential energy into kinetic energy鈥攖hat is, speed.

The consistency of "g" is what makes calculations like the one we described possible. In the equation used in the solution, the height of the waterfall directly influences the speed鈥攂ecause the longer the fall (meaning the more time the force of gravity has to act on the object), the faster it can go.
  • Constant acceleration = 9.8 m/s虏
  • Key in converting potential energy to kinetic energy
  • Downward force acting on all objects

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Most popular questions from this chapter

The speeder passes the position of the police car with a constant speed of \(15 \mathrm{m} / \mathrm{s}\). The police car immediately starts from rest and pursues the speeder with constant acceleration. What acceleration must the police car have if it is to catch the speeder in 7.0 s? Measure time from the moment the police car starts.

Bill steps off a 3.0 -m-high diving board and drops to the water below. At the same time, Ted jumps upward with a speed of \(4.2 \mathrm{m} / \mathrm{s}\) from a \(1.0-\mathrm{m}\) -high diving board. Choosing the origin to be at the water's surface, and upward to be the positive \(x\) direction, write \(x\) -versus-t equations of motion for both Bill and Ted.

Landing with a speed of \(81.9 \mathrm{m} / \mathrm{s}\), and traveling due south, a jet comes to rest in 949 m. Assuming the jet slows with constant acceleration, find the magnitude and direction of its acceleration.

Jules Verne In his novel From the Earth to the Moon (1866). Jules Verne describes a spaceship that is blasted out of a cannon, called the Columbiad, with a speed of 12,000 yards/s. The Columliad is \(900 \mathrm{ft}\) long. but part of it is packed with powder, so the spaceship accelerates over a distance of only 700 ft. Fstimate the acceleration experienced by the occupants of the spaceship during launch. Give your answer in \(\mathrm{m} / \mathrm{s}^{2}\). (Verne realized that the "travelers would. ... encounter a violent recoil, " but he probably didn't know that people generally lose consciousness if they experience accelerations greater than about \(7 g \sim 70 \mathrm{m} / \mathrm{s}^{2}\) )

The position of a particle as a function of time is given by \(x=(6 \mathrm{m} / \mathrm{s}) t+\left(-2 \mathrm{m} / \mathrm{s}^{2}\right) t^{2}\) (a) Plot \(x\) versus \(t\) for \(t=0\) to \(t=2 \mathrm{s} .\) (b) Find the average velocity of the particle from \(f=0\) to \(t=1 \mathrm{s}\). (c) Find the average speed from \(t=0\) to \(t=1 \mathrm{s}\).
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