A cosmic-ray proton entering the atmosphere from space has a...

A cosmic-ray proton entering the atmosphere from space has a kinetic energy of $2.0 \times 10^{20} \mathrm{eV} .$ (a) What is its kinetic energy in joules? (b) If all of the kinetic energy of the proton could be harnessed to lift an object of mass $1.0 \mathrm{kg}$ near Earth's surface, how far could the object be lifted? (c) What is the speed of the proton? $\left[\right.$Hint: Note that $\left.K \gg E_{0 \cdot}\right]$

A cosmic-ray proton entering the atmosphere from space has a kinetic energy of $2.0 \times 10^{20} \mathrm{eV} .$ (a) What is its kinetic energy in joules? (b) If all of the kinetic energy of the proton could be harnessed to lift an object of mass $1.0 \mathrm{kg}$ near Earth's surface, how far could the object be lifted?
(c) What is the speed of the proton? $\left[\right.$Hint: Note that $\left.K \gg E_{0 \cdot}\right]$

Key Concepts

Relativistic Kinetic Energy

When dealing with particles at speeds approaching the speed of light, classical mechanics is insufficient, and relativistic mechanics must be applied. Relativistic kinetic energy is given by the equation K = (? - 1)mc², where ? is the Lorentz factor and m is the rest mass. Especially when the kinetic energy is much greater than the rest mass energy (K >> mc²), the speed of the particle approaches the speed of light, necessitating a relativistic treatment.

Electron Volt to Joule Conversion

This concept involves converting energy expressed in electron volts (eV) to the standard SI unit of energy, the joule (J). The conversion factor is based on the definition that 1 eV is approximately 1.602 x 10^(-19) joules. This is crucial when handling high-energy particles and making further calculations in SI units.

Gravitational Potential Energy

Gravitational potential energy refers to the energy an object possesses due to its position in a gravitational field. It is typically calculated using the equation U = mgh, where m is mass, g is the acceleration due to gravity, and h is the height. This concept is used to determine the height to which an object can be raised if a specified amount of energy is available.

Transcript

00:01 So given in the question the kinetic energy of the cosmic ray proton entering the atmosphere from space is 2 .0 into 10 to the power 20 electron volt.

00:18 Let it be represented by k.

00:24 So k is 2 .0 into 10 to the power 20 electron volts and it has been asked to find its kinetic energy in joules.

00:33 In the second part if all the kinetic energy of the proton could be harnessed to lift an object of mass 1 .0 kgs near earth's surface the height up to which the object could be lifted and the third is the speed of the proton.

01:11 The third part is the speed of the proton.

01:18 Now we know that 1 electron volt is 1 .6 into 10 to the power minus 19 joules.

01:26 That means 2 .0 into 10 to the power 20 electron volt would be 2 .0 into 10 to the power 20 electron volt times 1 .6 into 10 to the power minus 19 joules per electron volt.

01:36 That comes out to be 32 joules.

01:39 Hence the kinetic energy of the cosmic ray proton is 32 joules.

01:48 Coming to the second part of the question let the height up to which the object could be lifted be h.

02:00 While lifting the object the entire kinetic energy of the proton will be converted into the potential energy of the object.

02:14 The potential energy of the object is mgh.

02:17 The kinetic energy of the proton is k.

02:20 So mgh will be equal to k which is equal to 32 joules...

Please give Ace some feedback