So, how does this play into things that we can measure like momentum? Well we remember what momentum was, according to Newton's model, which we can call classical, P is just M times V. Okay, if you double V, you double the amount of momentum, but you can't just keep doubling V forever and ever because there's this universal speed limit to the world that we live in, which is C. All right, you can't take V up above C, something bad happens when you do that. And so we need the relativistic version of momentum, which looks almost the same except we're gonna stick a gamma right in front of it. Alright, and again, gamma is equal to 1 over the square root, 1 minus V squared over C squared. Okay, it's a number bigger than 1. So relativistic momentum has more momentum in it than classical momentum. All right, so let's think about this now in terms of one of your homework problems and this is -- let's see if I can figure out what number it is. Okay, this is one that we looked at earlier in office hours, it is 26.18 and it says the following. Let particle of mass M move at some V, we'll call it V1 of 0.37 C. How fast does V2 have to be to have twice the momentum? Okay, so particle moving at V1, it's got some momentum, we want to go faster to V2 and we want to hit twice the momentum. So, if you were in the classical world obviously you would just double it, right? Double V, it doubles the momentum. But now, in this relativistic world we have this factor gamma which has V in it. All right, so we have two relationships, we have P1 is equal to gamma 1 mV1, and then we have P2, which is equal to gamma 2mV2, and we know that that is going to be twice the initial momentum P1. All right, so now we have this relationship here and we need to solve for V2. All right, so let's see if we can do that. Alright, so let's write it over here we've got two times P1 is equal to gamma 2mV2, and we know what P1 is it's gamma 1mV1. Alright, right off the bat we can get rid of m. Okay, m is the rest mass, they were calling it M naught, it's the same thing, M is the rest mass. Those cancel out -- those cancel out and now let's do a little trick, let's divide by gamma 2 and let's divide by gamma 1 on the right side, okay? So, I took gamma 1 and put it under there, I took gamma 2 and put it under there, and now let's square both sides. So if I square this and I square this, what do we get? We get 4 V1 squared over gamma 2 squared, equals V2 squared over gamma 1 squared. Okay, but we know that gamma 2 squared is 1 over the square root of 1 minus V squared over C squared, so 1 over gamma 2 squared has to be 1 minus V2 squared over C squared. 1 over gamma 1 squared is going to be 1 minus V1 squared over C squared. And so now this whole thing here becomes the following, we get 4 V1 squared times 1 over gamma 2 squared, which is 1 minus V2 squared over C squared, and on the right side we have V2 squared times 1 over gamma 1 squared, which is 1 minus V1 squared over C squared. And now we can just simplify this and solve for V2. All right, let's just do it real quick. Multiply everything by C squared we get 4 V1 squared times C squared minus 4 V1 squared V2 squared, equals V2 squared C squared minus V2 squared V1 squared, and now I have a V1 squared, a 2 V2 squared -- I have a V1 squared, a 2 V2 squared and so I can put those together, and then I have 4 V1 squared C squared that's left. When I add 4 to that negative 1, I get 3 V1 squared V2 squared and now we can put the V2's together, V2 squared times C squared plus 3 V1 squared and then finally we can divide by that quantity and you end up with the following. V2 is equal to -- and I'm going to take the square root, so I can do that right now, 2 V1 C, all over the square root of C squared plus 3 V1 squared. Okay, I went through that slightly quickly but you can replay the video later on and double-check your algebra and make sure you get the same thing, and now you have all these numbers, right? We have V1 we have C obviously and if you plug in all those numbers, you can double check with mine, you should get 0.62, and it wants it in the units of speed of light, and so you write 0.62 and then the units are C. Okay, so those were for my numbers, you of course will have different numbers for your V1. That one look familiar Jessica? Yeah. That's the one we went over earlier in office hours, painfully, I'm a add. All right, questions about this one before -- before we move on?
Table of contents
- 0. Math Review31m
- 1. Intro to Physics Units1h 23m
- 2. 1D Motion / Kinematics3h 56m
- Vectors, Scalars, & Displacement13m
- Average Velocity32m
- Intro to Acceleration7m
- Position-Time Graphs & Velocity26m
- Conceptual Problems with Position-Time Graphs22m
- Velocity-Time Graphs & Acceleration5m
- Calculating Displacement from Velocity-Time Graphs15m
- Conceptual Problems with Velocity-Time Graphs10m
- Calculating Change in Velocity from Acceleration-Time Graphs10m
- Graphing Position, Velocity, and Acceleration Graphs11m
- Kinematics Equations37m
- Vertical Motion and Free Fall19m
- Catch/Overtake Problems23m
- 3. Vectors2h 43m
- Review of Vectors vs. Scalars1m
- Introduction to Vectors7m
- Adding Vectors Graphically22m
- Vector Composition & Decomposition11m
- Adding Vectors by Components13m
- Trig Review24m
- Unit Vectors15m
- Introduction to Dot Product (Scalar Product)12m
- Calculating Dot Product Using Components12m
- Intro to Cross Product (Vector Product)23m
- Calculating Cross Product Using Components17m
- 4. 2D Kinematics1h 42m
- 5. Projectile Motion3h 6m
- 6. Intro to Forces (Dynamics)3h 22m
- 7. Friction, Inclines, Systems2h 44m
- 8. Centripetal Forces & Gravitation7h 26m
- Uniform Circular Motion7m
- Period and Frequency in Uniform Circular Motion20m
- Centripetal Forces15m
- Vertical Centripetal Forces10m
- Flat Curves9m
- Banked Curves10m
- Newton's Law of Gravity30m
- Gravitational Forces in 2D25m
- Acceleration Due to Gravity13m
- Satellite Motion: Intro5m
- Satellite Motion: Speed & Period35m
- Geosynchronous Orbits15m
- Overview of Kepler's Laws5m
- Kepler's First Law11m
- Kepler's Third Law16m
- Kepler's Third Law for Elliptical Orbits15m
- Gravitational Potential Energy21m
- Gravitational Potential Energy for Systems of Masses17m
- Escape Velocity21m
- Energy of Circular Orbits23m
- Energy of Elliptical Orbits36m
- Black Holes16m
- Gravitational Force Inside the Earth13m
- Mass Distribution with Calculus45m
- 9. Work & Energy1h 59m
- 10. Conservation of Energy2h 54m
- Intro to Energy Types3m
- Gravitational Potential Energy10m
- Intro to Conservation of Energy32m
- Energy with Non-Conservative Forces20m
- Springs & Elastic Potential Energy19m
- Solving Projectile Motion Using Energy13m
- Motion Along Curved Paths4m
- Rollercoaster Problems13m
- Pendulum Problems13m
- Energy in Connected Objects (Systems)24m
- Force & Potential Energy18m
- 11. Momentum & Impulse3h 40m
- Intro to Momentum11m
- Intro to Impulse14m
- Impulse with Variable Forces12m
- Intro to Conservation of Momentum17m
- Push-Away Problems19m
- Types of Collisions4m
- Completely Inelastic Collisions28m
- Adding Mass to a Moving System8m
- Collisions & Motion (Momentum & Energy)26m
- Ballistic Pendulum14m
- Collisions with Springs13m
- Elastic Collisions24m
- How to Identify the Type of Collision9m
- Intro to Center of Mass15m
- 12. Rotational Kinematics2h 59m
- 13. Rotational Inertia & Energy7h 4m
- More Conservation of Energy Problems54m
- Conservation of Energy in Rolling Motion45m
- Parallel Axis Theorem13m
- Intro to Moment of Inertia28m
- Moment of Inertia via Integration18m
- Moment of Inertia of Systems23m
- Moment of Inertia & Mass Distribution10m
- Intro to Rotational Kinetic Energy16m
- Energy of Rolling Motion18m
- Types of Motion & Energy24m
- Conservation of Energy with Rotation35m
- Torque with Kinematic Equations56m
- Rotational Dynamics with Two Motions50m
- Rotational Dynamics of Rolling Motion27m
- 14. Torque & Rotational Dynamics2h 5m
- 15. Rotational Equilibrium3h 39m
- 16. Angular Momentum3h 6m
- Opening/Closing Arms on Rotating Stool18m
- Conservation of Angular Momentum46m
- Angular Momentum & Newton's Second Law10m
- Intro to Angular Collisions15m
- Jumping Into/Out of Moving Disc23m
- Spinning on String of Variable Length20m
- Angular Collisions with Linear Motion8m
- Intro to Angular Momentum15m
- Angular Momentum of a Point Mass21m
- Angular Momentum of Objects in Linear Motion7m
- 17. Periodic Motion2h 9m
- 18. Waves & Sound3h 40m
- Intro to Waves11m
- Velocity of Transverse Waves21m
- Velocity of Longitudinal Waves11m
- Wave Functions31m
- Phase Constant14m
- Average Power of Waves on Strings10m
- Wave Intensity19m
- Sound Intensity13m
- Wave Interference8m
- Superposition of Wave Functions3m
- Standing Waves30m
- Standing Wave Functions14m
- Standing Sound Waves12m
- Beats8m
- The Doppler Effect7m
- 19. Fluid Mechanics2h 27m
- 20. Heat and Temperature3h 7m
- Temperature16m
- Linear Thermal Expansion14m
- Volume Thermal Expansion14m
- Moles and Avogadro's Number14m
- Specific Heat & Temperature Changes12m
- Latent Heat & Phase Changes16m
- Intro to Calorimetry21m
- Calorimetry with Temperature and Phase Changes15m
- Advanced Calorimetry: Equilibrium Temperature with Phase Changes9m
- Phase Diagrams, Triple Points and Critical Points6m
- Heat Transfer44m
- 21. Kinetic Theory of Ideal Gases1h 50m
- 22. The First Law of Thermodynamics1h 26m
- 23. The Second Law of Thermodynamics3h 11m
- 24. Electric Force & Field; Gauss' Law3h 42m
- 25. Electric Potential1h 51m
- 26. Capacitors & Dielectrics2h 2m
- 27. Resistors & DC Circuits3h 8m
- 28. Magnetic Fields and Forces2h 23m
- 29. Sources of Magnetic Field2h 30m
- Magnetic Field Produced by Moving Charges10m
- Magnetic Field Produced by Straight Currents27m
- Magnetic Force Between Parallel Currents12m
- Magnetic Force Between Two Moving Charges9m
- Magnetic Field Produced by Loops and Solenoids42m
- Toroidal Solenoids aka Toroids12m
- Biot-Savart Law (Calculus)18m
- Ampere's Law (Calculus)17m
- 30. Induction and Inductance3h 37m
- 31. Alternating Current2h 37m
- Alternating Voltages and Currents18m
- RMS Current and Voltage9m
- Phasors20m
- Resistors in AC Circuits9m
- Phasors for Resistors7m
- Capacitors in AC Circuits16m
- Phasors for Capacitors8m
- Inductors in AC Circuits13m
- Phasors for Inductors7m
- Impedance in AC Circuits18m
- Series LRC Circuits11m
- Resonance in Series LRC Circuits10m
- Power in AC Circuits5m
- 32. Electromagnetic Waves2h 14m
- 33. Geometric Optics2h 57m
- 34. Wave Optics1h 15m
- 35. Special Relativity2h 10m
35. Special Relativity
Consequences of Relativity
Video duration:
8mPlay a video:
Related Videos
Related Practice