Physics Unit 12: Special Relativity

1. Define inertial reference frame, proper time, dilated time, proper length, contracted length, relativistic momentum, nonrelativistic momentum, 2. Know the relativity postulates and their consequences. 3. An astronaut travels at 1x108 m/s for 24 hours as measured by ground control. What is the time as measured by the astronaut? 4. An alien flies by a football game at 0.90c and measures the time it takes to kick a field goal as 0.50 s. What is the proper time for the kick? 5. A meter stick is measured to be 50 cm long. How fast must the meter stick be traveling? 6. What is the relativistic momentum of an electron traveling at 0.99c? 7. A car is 500 kg at rest. What is the increase in its energy when it is traveling at 0.90c? 8. How much energy will be released when 2 kg of pencil is converted to energy? 9. What is the ratio of relativistic kinetic energy to classical kinetic energy for a 500 kg car traveling at 0.90c? 10. The Enterprise moves at 0.9c relative to earth and the Klingon Bird-of-Prey moves at 0.7c relative to earth. What does the navigator of the Bird-of-Prey report for the speed of the Enterprise? 11. The Klingon Battle Cruiser moving at 0.7c relative to earth fires a torpedo at 0.5c relative to the Battle Cruiser. What is the speed of the torpedo as observed from earth? 12. The Klingon Battle Cruise approaches earth at 0.7c relative to earth. It passes the Ferengi Shuttle at 0.5c relative to the shuttle. What is the speed of the Shuttle relative to the earth? 13. The starship Enterprise approaches the planet Risa at a speed of 0.5c relative to the planet. On the way, it overtakes the intergalactic freighter Astra. The relative speed of the two ships as measured by the navigator on the Enterprise is 0.1c. If the Astra has a red ( = 650 nm) navigation light, what wavelength will the Enterprise see as they approach the Astra.

8 푚 2 3. 훥푡 = 24 ℎ, 푣 = 1 × 10 , 훥푡 = ? 8 푚 푠 0 퐸 = (2 푘𝑔) (3 × 10 ) 푠 훥푡 훥푡 = 0 푬 = ퟏ. ퟖ × ퟏퟎퟏퟕ 푱 푣2 1 √1 − 9. 퐾퐸 = 푚푣2 푐2 푐푙푎푠푠푖푐 2 2 훥푡 1 푚 0 퐾퐸 = (500 푘𝑔) (0.90 (3 × 108 )) 24 ℎ = 푐푙푎푠푠푖푐 2 푠 푚 2 (1 × 108 ) 퐾퐸 = 1.82 × 1019 퐽 √1 − 푠 푐푙푎푠푠푖푐 푚 2 19 (3 × 108 ) 퐾퐸푟푒푙푎푡푖푣푖푠푡푖푐 = 5.82 × 10 퐽 (see #7) 푠 5.82 × 1019 퐽 훥푡 푟푎푡𝑖표 = = ퟑ. ퟐퟎ 24 ℎ = 0 1.82 × 1019 퐽 1 √1 − 10. 푣퐸푛푡퐸 = 0.9푐, 푣퐾퐸 = 0.7푐, 푣퐸푛푡퐾 = ? 9 푣퐸푛푡퐸 + 푣퐸퐾 8 푣퐸푛푡퐾 = 푣 푣 24 ℎ √ = 훥푡 1 + 퐸푛푡퐸 퐸퐾 9 0 푐2 0.9푐 + −0.7푐 휟풕ퟎ = ퟐퟐ. ퟔ 풉 푣 = 퐸푛푡퐾 (0.9푐)(−0.7푐) 4. 푣 = 0.90푐, 훥푡 = 0.50 푠, 훥푡0 = ? 1 + 푐2 훥푡0 0.50 푠 = 풗푬풏풕푲 = ퟎ. ퟓퟒퟏ풄 (0.90푐)2 √1 − 11. 푣 = 0.7푐, 푣 = 0.5푐, 푣 = ? 푐2 퐾퐸 푇퐾 푡퐸 푣푇퐾 + 푣퐾퐸 2 푣푇퐸 = 0.50 푠 √1 − 0.90 = 훥푡0 푣푇퐾푣퐾퐸 1 + 2 휟풕ퟎ = ퟎ. ퟐퟏퟖ 풔 푐 0.5푐 + 0.7푐 5. 퐿0 = 1 푚, 퐿 = 0.5 푚, 푣 = ? 푣 = 푇퐸 (0.7푐)(0.5푐) 푣2 1 + 퐿 = 퐿 √1 − 푐2 0 푐2 풗푻푬 = ퟎ. ퟖퟖퟗ풄 2 푣 12. 푣 = 0.7푐, 푣 = 0.5푐, 푣 = ? 0.5 푚 = 1 푚 √1 − 2 퐾퐸 퐾푆 푆퐸 푐 푣 + 푣 2 푆퐾 퐾퐸 푣 푣푆퐸 = 0.25 = 1 − 푣푆퐾푣퐾퐸 푐2 1 + 푐2 푣2 0.75 = −0.5푐 + 0.7푐 푐2 푣 = 푆퐸 (−0.5푐)(0.7푐) 푣 = √0.75푐 1 + 푐2 풗 = ퟎ. ퟖퟕ풄 풗푺푬 = ퟎ. ퟑퟎퟖ풄 −31 6. 푣 = 0.99푐, 푚 = 9.11 × 10 푘𝑔, 푝 = ? 13. 푢 = −0.1푐, 휆 = 650 푛푚 푚푣 푠 푝 = 2 √ 푣 푢 1− 2 1 + 푐 푐 푚 휆표푏푠 = 휆푠√ 푢 (9.11 × 10−31 푘𝑔) (0.99 (3 × 108 )) 1 − 푠 푐 푝 = (0.99푐)2 √1 − −0.1푐 푐2 1 + 휆 = (650 푛푚)√ 푐 = ퟓퟖퟖ 풏풎 풑 = ퟏ. ퟗퟐ × ퟏퟎ−ퟐퟏ 풌품 풎/풔 표푏푠 −0.1푐 1 − 7. 푚 = 500 푘𝑔, 푣 = 0.90푐, 퐾퐸 = ? 푐

1 퐾퐸 = 푚푐2 − 1 푣2 √1 − ( 푐2 )

1 퐾퐸 = (500 푘𝑔)(푐2) − 1 0.90푐2 √1 − ( 푐2 ) 푲푬 = ퟓ. ퟖퟐ × ퟏퟎퟏퟗ 푱 8. 푚 = 2 푘𝑔, 퐸 = ? 퐸 = 푚푐2