what is the ratio of the kinetic energy of particle b to the kinetic energy of particle a?

7 Work and Kinetic Free energy

seven.2 Kinetic Energy

Learning Objectives

By the end of this department, y'all will be able to:

Calculate the kinetic energy of a particle given its mass and its velocity or momentum
Evaluate the kinetic energy of a body, relative to different frames of reference

It's plausible to suppose that the greater the velocity of a body, the greater consequence it could have on other bodies. This does not depend on the management of the velocity, only its magnitude. At the cease of the seventeenth century, a quantity was introduced into mechanics to explain collisions between two perfectly elastic bodies, in which one body makes a head-on standoff with an identical torso at remainder. The starting time body stops, and the second torso moves off with the initial velocity of the starting time body. (If yous have always played billiards or croquet, or seen a model of Newton'due south Cradle, y'all take observed this type of standoff.) The idea backside this quantity was related to the forces acting on a body and was referred to equally "the energy of motion." Later on, during the eighteenth century, the name kinetic energy was given to energy of motion.

With this history in mind, we can now state the classical definition of kinetic free energy. Note that when we say "classical," we mean non-relativistic, that is, at speeds much less that the speed of light. At speeds comparable to the speed of light, the special theory of relativity requires a unlike expression for the kinetic energy of a particle, as discussed in Relativity in the third volume of this text.

Since objects (or systems) of interest vary in complication, we first define the kinetic energy of a particle with mass grand.

Kinetic Energy

The kinetic free energy of a particle is one-half the production of the particle's mass m and the square of its speed v:

$K=\frac{1}{2}m{v}^{2}.$

We and then extend this definition to whatsoever organisation of particles past adding up the kinetic energies of all the elective particles:

$K=\sum \frac{1}{2}m{v}^{2}.$

Annotation that simply as we tin can limited Newton's second law in terms of either the rate of change of momentum or mass times the charge per unit of change of velocity, so the kinetic free energy of a particle can be expressed in terms of its mass and momentum

$(\overset{\to }{p}=m\overset{\to }{v}),$

instead of its mass and velocity. Since

$v=p\text{/}m$

, nosotros see that

$K=\frac{1}{2}m{(\frac{p}{m})}^{2}=\frac{{p}^{2}}{2m}$

also expresses the kinetic energy of a single particle. Sometimes, this expression is more convenient to use than (Figure).

The units of kinetic energy are mass times the square of speed, or

$\text{kg}·{\text{m}}^{2}{\text{/s}}^{2}$

. Simply the units of forcefulness are mass times acceleration,

$\text{kg}·{\text{m/s}}^{2}$

, so the units of kinetic energy are besides the units of force times distance, which are the units of work, or joules. You will see in the next section that work and kinetic energy take the same units, considering they are unlike forms of the same, more full general, concrete holding.

Example

Kinetic Free energy of an Object

(a) What is the kinetic energy of an 80-kg athlete, running at 10 k/s? (b) The Chicxulub crater in Yucatan, one of the largest existing impact craters on World, is thought to accept been created by an asteroid, traveling at

22 km/south and releasing

$4.2\,×\,{10}^{23}\,\text{J}$

of kinetic energy upon impact. What was its mass? (c) In nuclear reactors, thermal neutrons, traveling at about two.2 km/s, play an of import role. What is the kinetic energy of such a particle?

Strategy

To answer these questions, you can utilize the definition of kinetic energy in (Effigy). Y'all also have to look up the mass of a neutron.

Solution

Don't forget to convert km into grand to do these calculations, although, to save space, nosotros omitted showing these conversions.

$K=\frac{1}{2}(80\,\text{kg})(10\,{\text{m/s})}^{2}=4.0\,\text{kJ}\text{.}$
$m=2K\text{/}{v}^{2}=2(4.2\,×\,{10}^{23}\text{J})\text{/}{(22\,\text{km/s})}^{2}=1.7\,×\,{10}^{15}\,\text{kg}\text{.}$
$K=\frac{1}{2}(1.68\,×\,{10}^{-27}\,\text{kg}){(2.2\,\text{km/s})}^{2}=4.1\,×\,{10}^{-21}\,\text{J}\text{.}$

Significance

In this example, nosotros used the way mass and speed are related to kinetic energy, and we encountered a very broad range of values for the kinetic energies. Unlike units are ordinarily used for such very large and very small values. The free energy of the impactor in part (b) can be compared to the explosive yield of TNT and nuclear explosions,

$1\,\text{megaton}=4.18\,×\,{10}^{15}\,\text{J}\text{.}$

The Chicxulub asteroid's kinetic energy was about a hundred million megatons. At the other extreme, the free energy of subatomic particle is expressed in electron-volts,

$1\,\text{eV}=1.6\,×\,{10}^{-19}\,\text{J}\text{.}$

The thermal neutron in part (c) has a kinetic energy of about ane fortieth of an electron-volt.

Bank check Your Understanding

(a) A auto and a truck are each moving with the same kinetic free energy. Presume that the truck has more mass than the car. Which has the greater speed? (b) A car and a truck are each moving with the same speed. Which has the greater kinetic free energy?

[reveal-answer q="119029″]Show Solution[/reveal-answer]
[subconscious-answer a="119029″]a. the car; b. the truck[/hidden-answer]

Because velocity is a relative quantity, yous tin see that the value of kinetic energy must depend on your frame of reference. Yous can by and large choose a frame of reference that is suited to the purpose of your analysis and that simplifies your calculations. One such frame of reference is the 1 in which the observations of the organisation are made (probable an external frame). Some other choice is a frame that is attached to, or moves with, the system (likely an internal frame). The equations for relative movement, discussed in Motion in Two and 3 Dimensions, provide a link to calculating the kinetic free energy of an object with respect to unlike frames of reference.

Instance

Kinetic Energy Relative to Different Frames

A 75.0-kg person walks downwards the fundamental alley of a subway auto at a speed of one.50 m/s relative to the car, whereas the train is moving at xv.0 m/s relative to the tracks. (a) What is the person's kinetic energy relative to the machine? (b) What is the person'south kinetic energy relative to the tracks? (c) What is the person's kinetic free energy relative to a frame moving with the person?

Strategy

Since speeds are given, we can apply

$\frac{1}{2}m{v}^{2}$

to calculate the person's kinetic energy. All the same, in part (a), the person's speed is relative to the subway motorcar (equally given); in role (b), it is relative to the tracks; and in part (c), information technology is nada. If we denote the car frame by C, the track frame by T, and the person past P, the relative velocities in part (b) are related by

${\overset{\to }{v}}_{\text{PT}}={\overset{\to }{v}}_{\text{PC}}+{\overset{\to }{v}}_{\text{CT}}.$

We tin assume that the central alley and the tracks lie forth the same line, but the management the person is walking relative to the car isn't specified, then we will give an answer for each possibility,

${v}_{\text{PT}}={v}_{\text{CT}}±{v}_{\text{PC}}$

, as shown in (Effigy).

Two illustrations of a person walking in a train car. In figure a, the person is moving to the right with velocity vector v sub P C and the train is moving to the right with velocity vector v sub C T. In figure b, the person is moving to the left with velocity vector v sub P C and the train is moving to the right with velocity vector v sub C T. — **Figure 7.10** The possible motions of a person walking in a train are (a) toward the front end of the automobile and (b) toward the dorsum of the machine.

Solution

$K=\frac{1}{2}(75.0\,\text{kg})(1.50\,{\text{m/s})}^{2}=84.4\,\text{J}\text{.}$
${v}_{\text{PT}}=(15.0±1.50)\,\text{m/s}\text{.}$

Therefore, the 2 possible values for kinetic free energy relative to the car are

$K=\frac{1}{2}(75.0\,\text{kg})(13.5\,{\text{m/s})}^{2}=6.83\,\text{kJ}$

and

$K=\frac{1}{2}(75.0\,\text{kg})(16.5\,{\text{m/s})}^{2}=10.2\,\text{kJ}\text{.}$
In a frame where
${v}_{\text{P}}=0,K=0$

as well.

Significance

You lot tin see that the kinetic energy of an object can have very different values, depending on the frame of reference. All the same, the kinetic energy of an object can never be negative, since information technology is the product of the mass and the square of the speed, both of which are ever positive or naught.

Check Your Understanding

You lot are rowing a boat parallel to the banks of a river. Your kinetic energy relative to the banks is less than your kinetic energy relative to the h2o. Are you rowing with or against the electric current?

[reveal-answer q="482834″]Evidence Solution[/reveal-answer]
[hidden-reply a="482834″]against[/hidden-answer]

The kinetic energy of a particle is a single quantity, but the kinetic energy of a organisation of particles tin can sometimes be divided into diverse types, depending on the system and its motion. For example, if all the particles in a system have the same velocity, the system is undergoing translational motion and has translational kinetic free energy. If an object is rotating, it could have rotational kinetic free energy, or if it'due south vibrating, information technology could take vibrational kinetic free energy. The kinetic energy of a system, relative to an internal frame of reference, may be called internal kinetic free energy. The kinetic energy associated with random molecular motion may exist called thermal energy. These names will be used in later capacity of the book, when appropriate. Regardless of the name, every kind of kinetic energy is the same physical quantity, representing energy associated with motion.

Example

Special Names for Kinetic Energy

(a) A thespian lobs a mid-court pass with a 624-k basketball, which covers 15 m in 2 s. What is the basketball's horizontal translational kinetic energy while in flight? (b) An average molecule of air, in the basketball in part (a), has a mass of 29 u, and an average speed of 500 1000/s, relative to the basketball game. There are well-nigh

$3\,×\,{10}^{23}$

molecules inside it, moving in random directions, when the ball is properly inflated. What is the average translational kinetic energy of the random motion of all the molecules within, relative to the basketball? (c) How fast would the basketball have to travel relative to the courtroom, every bit in part (a), so every bit to have a kinetic free energy equal to the amount in part (b)?

Strategy

In part (a), first find the horizontal speed of the basketball and so use the definition of kinetic free energy in terms of mass and speed,

$K=\frac{1}{2}m{v}^{2}$

. Then in part (b), convert unified units to kilograms and so employ

$K=\frac{1}{2}m{v}^{2}$

to go the average translational kinetic energy of one molecule, relative to the basketball game. So multiply by the number of molecules to go the total effect. Finally, in part (c), we can substitute the amount of kinetic energy in part (b), and the mass of the basketball in part (a), into the definition

$K=\frac{1}{2}m{v}^{2}$

, and solve for 5.

Solution

The horizontal speed is (15 m)/(2 s), then the horizontal kinetic energy of the basketball is
$\frac{1}{2}(0.624\,\text{kg}){(7.5\,\text{m/s})}^{2}=17.6\,\text{J}\text{.}$
The average translational kinetic free energy of a molecule is
$\frac{1}{2}(29\,\text{u})(1.66\,×\,{10}^{-27}\,\text{kg/u}){(500\,\text{m/s})}^{2}=6.02\,×\,{10}^{-21}\,\text{J,}$

and the total kinetic free energy of all the molecules is

$(3\,×\,{10}^{23})(6.02\,×\,{10}^{-21}\,\text{J})=1.80\,\text{kJ}\text{.}$
$v=\sqrt{2(1.8\,\text{kJ})\text{/}(0.624\,\text{kg})}=76.0\,\text{m/s}\text{.}$

Significance

In role (a), this kind of kinetic energy tin be called the horizontal kinetic free energy of an object (the basketball), relative to its surroundings (the court). If the basketball were spinning, all parts of it would take not just the boilerplate speed, simply information technology would also have rotational kinetic free energy. Part (b) reminds us that this kind of kinetic energy can be called internal or thermal kinetic energy. Notice that this energy is about a hundred times the energy in part (a). How to make use of thermal energy volition be the subject field of the chapters on thermodynamics. In part (c), since the free energy in part (b) is about 100 times that in part (a), the speed should exist about 10 times as big, which information technology is (76 compared to seven.5 m/s).

Summary

The kinetic energy of a particle is the product of one-half its mass and the square of its speed, for not-relativistic speeds.
The kinetic free energy of a organization is the sum of the kinetic energies of all the particles in the system.
Kinetic free energy is relative to a frame of reference, is e'er positive, and is sometimes given special names for dissimilar types of motion.

Conceptual Questions

A particle of m has a velocity of

${v}_{x}\hat{i}+{v}_{y}\hat{j}+{v}_{z}\hat{k}.$

Is its kinetic free energy given past

$m({v}_{x}{}^{2}\hat{i}+{v}_{y}{}^{2}\hat{j}+{v}_{z}{}^{2}\hat{k}\text{)/2?}$

If not, what is the correct expression?

Ane particle has mass thou and a second particle has mass twom. The second particle is moving with speed five and the get-go with speed 25. How exercise their kinetic energies compare?

[reveal-answer q="fs-id1165036746350″]Show Solution[/reveal-answer]

[hidden-answer a="fs-id1165036746350″]

The get-go particle has a kinetic energy of

$4(\frac{1}{2}m{v}^{2})$

whereas the second particle has a kinetic energy of

$2(\frac{1}{2}m{v}^{2}),$

and so the first particle has twice the kinetic energy of the second particle.
[/hidden-answer]

A person drops a pebble of mass

${m}_{1}$

from a tiptop h, and it hits the flooring with kinetic energy Thou. The person drops some other pebble of mass

${m}_{2}$

from a elevation of 2h, and it hits the floor with the same kinetic energy K. How do the masses of the pebbles compare?

Problems

Compare the kinetic free energy of a 20,000-kg truck moving at 110 km/h with that of an fourscore.0-kg astronaut in orbit moving at 27,500 km/h.

(a) How fast must a 3000-kg elephant move to take the aforementioned kinetic energy as a 65.0-kg sprinter running at 10.0 k/s? (b) Discuss how the larger energies needed for the movement of larger animals would relate to metabolic rates.

[reveal-answer q="fs-id1165036994946″]Show Solution[/reveal-reply]

[hidden-answer a="fs-id1165036994946″]

a. 1.47 yard/s; b. answers may vary

[/hidden-answer]

Gauge the kinetic energy of a ninety,000-ton aircraft carrier moving at a speed of at 30 knots. You will need to look up the definition of a nautical mile to use in converting the unit for speed, where i knot equals i nautical mile per hour.

Calculate the kinetic energies of (a) a 2000.0-kg automobile moving at 100.0 km/h; (b) an 80.-kg runner sprinting at 10. m/s; and (c) a

$9.1\,×\,{10}^{-31}\,\text{-kg}$

electron moving at

$2.0\,×\,{10}^{7}\,\text{m/s}\text{.}$

[reveal-answer q="fs-id1165037853891″]Prove Solution[/reveal-respond]

[hidden-answer a="fs-id1165037853891″]

a. 772 kJ; b. 4.0 kJ; c.

$1.8\,×\,{10}^{-16}\,\text{J}$

[/hidden-answer]

A 5.0-kg torso has 3 times the kinetic energy of an 8.0-kg trunk. Calculate the ratio of the speeds of these bodies.

An 8.0-thou bullet has a speed of 800 thousand/s. (a) What is its kinetic energy? (b) What is its kinetic energy if the speed is halved?

[reveal-answer q="fs-id1165037053928″]Bear witness Solution[/reveal-respond]

[hidden-answer a="fs-id1165037053928″]

a. 2.half-dozen kJ; b. 640 J

[/hidden-answer]

Glossary

kinetic energy: free energy of motion, one-half an object'south mass times the foursquare of its speed

coaldrakekned1979.blogspot.com

Source: https://opentextbc.ca/universityphysicsv1openstax/chapter/7-2-kinetic-energy/

what is the ratio of the kinetic energy of particle b to the kinetic energy of particle a?

seven.2 Kinetic Energy

Learning Objectives

Kinetic Energy

Example

Kinetic Free energy of an Object

Strategy

Solution

Significance

Bank check Your Understanding

Instance

Kinetic Energy Relative to Different Frames

Strategy

Solution

Significance

Check Your Understanding

Example

Special Names for Kinetic Energy

Strategy

Solution

Significance

Summary

Conceptual Questions

Problems

Glossary

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