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Chapter 2: Q83E (page 67)

How fast must an object be moving for its kinetic energy to equal its internal energy?

Short Answer

Expert verified

The object must be moving at a speed of0.866c

Step by step solution

01

Define Internal energy

The total energy of an object which is stationary in an isolated system is,

Eint=mc2

Here, m is the mass and c is the speed of light.

02

Define Kinetic energy equation in relativistic terms

Kinetic energy is associated with the motion of the object It could be determined by substracting object鈥檚 energy (Internal) at rest from energy of moving object.

Therefore,

KE=energyofmotion-energyatrest=纬mc2-mc2=(纬-1)mc2

Here,纬istherelativisticfactor.

Equating both of the above questions and solving,

KE=Eint(纬-1)mc2=mc2纬mc2=2mc2纬=2

Write the equation for relativistic factor as given below.

纬=11-v2c22=11-v2c21-v2c2=14v2c2=1-0.25

v2=0.75c2v=0.866c

Hence, for an object鈥檚 kinetic energy to be equal to its internal energy, it must travel at a speed of 0.866times the speed of light.

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Most popular questions from this chapter

Classically, the net work done on an initially stationary object equals the final kinetic energy of the object. Verify that this also holds relativistically. Consider only one-dimension motion. It will be helpful to use the expression for p as a function of u in the following:

W=鈭�Fdx=鈭�dpdtdx=鈭�dxdtdp=鈭�udp

A projectile is a distance r from the center of a heavenly body and is heading directly away. Classically, if the sum of its kinetic and potential energies is positive, it will escape the gravitational pull of the body, but if negative, it cannot escape. Now imagine that the projectile is a pulse of light energy E. Since light has no internal energy ,E is also the kinetic energy of the light pulse. Suppose that the gravitational potential energy of the light pulse is given by Newton鈥檚 classical formula U=-(GMm/r), where M is the mass of the heavenly body and m is an 鈥渆ffective mass鈥� of the light pulse. Assume that this effective mass is given by m=E/c2.

Show that the critical radius for which light could not escape the gravitational pull of a heavenly body is within a factor of 2 of the Schwarzschild radius given in the chapter. (This kind of 鈥渟emiclassical鈥� approach to general relativity is sometimes useful but always vague. To be reliable, predictions must be based from beginning to end on the logical, but unfortunately complex, fundamental equations of general relativity.)

(a) Determine the Lorentz transformation matrix giving position and time in framefromS'those in framein the classical limitlocalid="1657533931071" v<<鈥�c. (b) Show that it yields equations (2-1).

Bob is on Earth. Anna is on a spacecraft moving away from Earth at 0.6c . At some point in Anna's outward travel, Bob fires a projectile loaded with supplies out to Anna's ship. Relative to Bob, the projectile moves at 0.8c . (a) How fast does the projectile move relative to Anna? (b) Bob also sends a light signal, " Greetings from Earth:' out to Anna's ship. How fast does the light signal move relative to Anna?

For the situation given in Exercise 22, find the Lorentz transformation matrix from Bob鈥檚 frame to Anna鈥檚 frame, then solve the problem via matrix multiplication.
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