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HomeStudy GuidesPhysics

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The Nature of Science and Physics

Kinematics

Introduction to One-Dimensional KinematicsDisplacementVectors, Scalars, and Coordinate SystemsTime, Velocity, and SpeedVideo: One-Dimensional KinematicsAccelerationMotion Equations for Constant Acceleration in One DimensionProblem-Solving Basics for One-Dimensional KinematicsFalling ObjectsGraphical Analysis of One-Dimensional Motion

Two-Dimensional Kinematics

Dynamics: Force and Newton's Laws of Motion

Introduction to Dynamics: Newton's Laws of MotionDevelopment of Force ConceptNewton's First Law of Motion: InertiaNewton's Second Law of Motion: Concept of a SystemNewton's Third Law of Motion: Symmetry in ForcesVideo: Newton's LawsNormal, Tension, and Other Examples of ForcesProblem-Solving StrategiesFurther Applications of Newton's Laws of MotionExtended Topic: The Four Basic Forces—An Introduction

Further Applications of Newton's Laws: Friction, Drag, and Elasticity

Uniform Circular Motion and Gravitation

Introduction to Uniform Circular Motion and GravitationRotation Angle and Angular VelocityCentripetal AccelerationCentripetal ForceFictitious Forces and Non-inertial Frames: The Coriolis ForceNewton's Universal Law of GravitationVideo: GravitationSatellites and Kepler's Laws: An Argument for Simplicity

Work, Energy, and Energy Resources

Introduction to Work, Energy, and Energy ResourcesWork: The Scientific DefinitionKinetic Energy and the Work-Energy TheoremGravitational Potential EnergyVideo: Potential and Kinetic EnergyConservative Forces and Potential EnergyNonconservative ForcesConservation of EnergyPowerWork, Energy, and Power in HumansWorld Energy Use

Linear Momentum and Collisions

Rotational Motion and Angular Momentum

Introduction to Rotational Motion and Angular MomentumAngular AccelerationKinematics of Rotational MotionVideo: Rotational MotionDynamics of Rotational Motion: Rotational InertiaRotational Kinetic Energy: Work and Energy RevisitedAngular Momentum and Its ConservationVideo: Angular MomentumCollisions of Extended Bodies in Two DimensionsGyroscopic Effects: Vector Aspects of Angular Momentum

Statics and Torque

Fluid Statics

Introduction to Fluid StaticsWhat Is a Fluid?DensityPressureVariation of Pressure with Depth in a FluidPascal's PrincipleGauge Pressure, Absolute Pressure, and Pressure MeasurementArchimedes' PrincipleVideo: BuoyancyCohesion and Adhesion in Liquids: Surface Tension and Capillary ActionPressures in the Body

Fluid Dynamics and Its Biological and Medical Applications

Introduction to Fluid Dynamics and Biological and Medical ApplicationsFlow Rate and Its Relation to VelocityBernoulli's EquationVideo: Fluid FlowThe Most General Applications of Bernoulli's EquationViscosity and Laminar Flow; Poiseuille's LawThe Onset of TurbulenceMotion of an Object in a Viscous FluidMolecular Transport Phenomena: Diffusion, Osmosis, and Related Processes

Temperature, Kinetic Theory, and the Gas Laws

Heat and Heat Transfer Methods

Thermodynamics

Introduction to ThermodynamicsThe First Law of ThermodynamicsThe First Law of Thermodynamics and Some Simple ProcessesIntroduction to the Second Law of Thermodynamics: Heat Engines and Their EfficiencyCarnot's Perfect Heat Engine: The Second Law of Thermodynamics RestatedApplications of Thermodynamics: Heat Pumps and RefrigeratorsEntropy and the Second Law of Thermodynamics: Disorder and the Unavailability of EnergyStatistical Interpretation of Entropy and the Second Law of Thermodynamics: The Underlying Explanation

Oscillatory Motion and Waves

Introduction to Oscillatory Motion and WavesHooke's Law: Stress and Strain RevisitedPeriod and Frequency in OscillationsSimple Harmonic Motion: A Special Periodic MotionVideo: Harmonic MotionThe Simple PendulumEnergy and the Simple Harmonic OscillatorUniform Circular Motion and Simple Harmonic MotionDamped Harmonic MotionForced Oscillations and ResonanceWavesSuperposition and InterferenceEnergy in Waves: Intensity

Physics of Hearing

- State Newton’s third law of motion.
- Explain the principle involved in propulsion of rockets and jet engines.
- Derive an expression for the acceleration of the rocket.
- Discuss the factors that affect the rocket’s acceleration.
- Describe the function of a space shuttle.

By calculating the change in momentum for the entire system over Δ

$\displaystyle{a}=\frac{v_{\text{e}}}{m}\frac{\Delta{m}}{\Delta{t}}-g\\$

“The rocket” is that part of the system remaining after the gas is ejected, and $\displaystyle{a}=\frac{v_{\text{e}}}{m}\frac{\Delta{m}}{\Delta{t}}-g\\$

where $\frac{\Delta{m}}{\Delta{t}}\\$

in the equation. The quantity $\left(\frac{\Delta{m}}{\Delta{t}}\right)v_{\text{e}}\\$

, with units of newtons, is called "thrust.” The faster the rocket burns its fuel, the greater its thrust, and the greater its acceleration. The third factor is the mass - The greater the exhaust velocity
*v*_{e}of the gases relative to the rocket, the greater the acceleration. - The faster the rocket burns its fuel, the greater its acceleration.
- The smaller the rocket’s mass (all other factors being the same), the greater the acceleration.

$\begin{array}{lll}a&=&\frac{v_{\text{e}}}{m}\frac{\Delta{m}}{\Delta{t}}-g\\\text{ }&=&\frac{2.40\times10^3\text{ m/s}}{2.80\times10^6\text{ kg}}\left(1.40\times10^4\text{ kg/s}\right)-9.80\text{ m/s}^2\\\text{ }&=&2.20\text{ m/s}^2\end{array}\\$

$\frac{\Delta{m}}{\Delta{t}}\\$

remain constant. Knowing this acceleration and the mass of the rocket, you can show that the thrust of the engines was 3.36 × 10$v=v_{\text{e}}\ln\frac{m_0}{m_\text{r}}\\$

, where $\ln\frac{m_0}{m_\text{r}}\\$

is the natural logarithm of the ratio of the initial mass of the rocket ($\displaystyle\ln\frac{m_0}{m_{\text{r}}}=\frac{v}{v_{\text{e}}}=\frac{11.2\times10^3\text{ m/s}}{2.5\times10^3\text{ m/s}}=4.48\\$

Solving for $\frac{m_0}{m_\text{r}}\\$

gives$\displaystyle\frac{m_0}{m_{\text{r}}}=e^{4.48}=88\\$

Thus, the mass of the rocket is$\displaystyle{m}_{\text{r}}=\frac{m_0}{88}\\$

This result means that only 1/88 of the mass is left when the fuel is burnt, and 87/88 of the initial mass was fuel. Expressed as percentages, 98.9% of the rocket is fuel, while payload, engines, fuel tanks, and other components make up only 1.10%. Taking air resistance and gravitational force into account, the mass

The space shuttle was an attempt at an economical vehicle with some reusable parts, such as the solid fuel boosters and the craft itself. (See Figure 2) The shuttle’s need to be operated by humans, however, made it at least as costly for launching satellites as expendable, unmanned rockets. Ideally, the shuttle would only have been used when human activities were required for the success of a mission, such as the repair of the Hubble space telescope. Rockets with satellites can also be launched from airplanes. Using airplanes has the double advantage that the initial velocity is significantly above zero and a rocket can avoid most of the atmosphere’s resistance.

- Newton’s third law of motion states that to every action, there is an equal and opposite reaction.
- Acceleration of a rocket is $\displaystyle{a}=\frac{v_{\text{e}}}{m}\frac{\Delta{m}}{\Delta{t}}-g\\$.
- A rocket’s acceleration depends on three main factors. They are

- The greater the exhaust velocity of the gases, the greater the acceleration.
- The faster the rocket burns its fuel, the greater its acceleration.
- The smaller the rocket's mass, the greater the acceleration.

**Professional Application.**Suppose a fireworks shell explodes, breaking into three large pieces for which air resistance is negligible. How is the motion of the center of mass affected by the explosion? How would it be affected if the pieces experienced significantly more air resistance than the intact shell?**Professional Application.**During a visit to the International Space Station, an astronaut was positioned motionless in the center of the station, out of reach of any solid object on which he could exert a force. Suggest a method by which he could move himself away from this position, and explain the physics involved.**Professional Application.**It is possible for the velocity of a rocket to be greater than the exhaust velocity of the gases it ejects. When that is the case, the gas velocity and gas momentum are in the same direction as that of the rocket. How is the rocket still able to obtain thrust by ejecting the gases?

**Professional Application.**Antiballistic missiles (ABMs) are designed to have very large accelerations so that they may intercept fast-moving incoming missiles in the short time available. What is the takeoff acceleration of a 10,000-kg ABM that expels 196 kg of gas per second at an exhaust velocity of 2.50 × 10^{3}m/s?**Professional Application.**What is the acceleration of a 5000-kg rocket taking off from the Moon, where the acceleration due to gravity is only 1.6 m/s^{2}, if the rocket expels 8.00 kg of gas per second at an exhaust velocity of 2.20 × 10^{3}m/s?**Professional Application.**Calculate the increase in velocity of a 4000-kg space probe that expels 3500 kg of its mass at an exhaust velocity of 2.00 × 10^{3}m/s. You may assume the gravitational force is negligible at the probe’s location.**Professional Application.**Ion-propulsion rockets have been proposed for use in space. They employ atomic ionization techniques and nuclear energy sources to produce extremely high exhaust velocities, perhaps as great as 8.00 × 10^{6}m/s. These techniques allow a much more favorable payload-to-fuel ratio. To illustrate this fact: (a) Calculate the increase in velocity of a 20,000-kg space probe that expels only 40.0-kg of its mass at the given exhaust velocity. (b) These engines are usually designed to produce a very small thrust for a very long time—the type of engine that might be useful on a trip to the outer planets, for example. Calculate the acceleration of such an engine if it expels 4.50 × 10^{−6}kg/s at the given velocity, assuming the acceleration due to gravity is negligible.- Derive the equation for the vertical acceleration of a rocket.
**Professional Application.**(a) Calculate the maximum rate at which a rocket can expel gases if its acceleration cannot exceed seven times that of gravity. The mass of the rocket just as it runs out of fuel is 75,000-kg, and its exhaust velocity is 2.40 × 10^{3}m/s. Assume that the acceleration of gravity is the same as on Earth’s surface (9.80 m/s^{2}). (b) Why might it be necessary to limit the acceleration of a rocket?- Given the following data for a fire extinguisher-toy wagon rocket experiment, calculate the average exhaust velocity of the gases expelled from the extinguisher. Starting from rest, the final velocity is 10.0 m/s. The total mass is initially 75.0 kg and is 70.0 kg after the extinguisher is fired.
- How much of a single-stage rocket that is 100,000 kg can be anything but fuel if the rocket is to have a final speed of 8.00km/s, given that it expels gases at an exhaust velocity of 2.20 × 10
^{3}m/s? **Professional Application.**(a) A 5.00-kg squid initially at rest ejects 0.250-kg of fluid with a velocity of 10.0 m/s. What is the recoil velocity of the squid if the ejection is done in 0.100 s and there is a 5.00-N frictional force opposing the squid’s movement. (b) How much energy is lost to work done against friction?**Unreasonable Results.**Squids have been reported to jump from the ocean and travel 30.0m (measured horizontally) before re-entering the water. (a) Calculate the initial speed of the squid if it leaves the water at an angle of 20.0º, assuming negligible lift from the air and negligible air resistance. (b) The squid propels itself by squirting water. What fraction of its mass would it have to eject in order to achieve the speed found in the previous part? The water is ejected at 12.0 m/s; gravitational force and friction are neglected. (c) What is unreasonable about the results? (d) Which premise is unreasonable, or which premises are inconsistent?**Construct Your Own Problem.**Consider an astronaut in deep space cut free from her space ship and needing to get back to it. The astronaut has a few packages that she can throw away to move herself toward the ship. Construct a problem in which you calculate the time it takes her to get back by throwing all the packages at one time compared to throwing them one at a time. Among the things to be considered are the masses involved, the force she can exert on the packages through some distance, and the distance to the ship.**Construct Your Own Problem.**Consider an artillery projectile striking armor plating. Construct a problem in which you find the force exerted by the projectile on the plate. Among the things to be considered are the mass and speed of the projectile and the distance over which its speed is reduced. Your instructor may also wish for you to consider the relative merits of depleted uranium versus lead projectiles based on the greater density of uranium.

3. 4.16 × 10

5. The force needed to give a small mass Δm an acceleration

$F=v_{\text{e}}\frac{\Delta{m}}{\Delta{t}}\\$

. By Newton’s third law, this force is equal in magnitude to the thrust force acting on the rocket, so $F_{\text{thrust}}=v_{\text{e}}\frac{\Delta{m}}{\Delta{t}}\\$

, where all quantities are positive. Applying Newton’s second law to the rocket gives $\displaystyle{a}=\frac{v_{\text{e}}}{m}\frac{\Delta{m}}{\Delta{t}}-g\\$

, where