Question
4. In experiments physicists routinely accelerate protons to speeds quite close to the speed of light. The mass of a proton is 1.67times 10^-27kg and the proton is moving with a speed of 0.99 c (a) Calculate the proton's momentum according to Newton's definition. (b) Calculate the proton's relativistic momentum. (c) Determine the ratio of the relativistic momentum to the Newtonian momentum. 5.The relativistic momentum of a particle of rest mass m and speed v is equal to 5mv.Calculate the speed of the particle.14. A An electron with a speed of 0.999c has a momentum that is equal to the momentum of a ship with a mass of 4.38times 10^7 kg moving at a certain speed Determine the speed of the ship. 7. If you were travelling on a spacecraft at 099c relative to Earth,would you feel compressed in the direction of travel? Explain your answer.Ku 8. Why do we not notice the effects of length contraction in our everyday lives?For example, why do cars not appear shorter when they drive past us at high speeds?
Solution
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Ivy
Professional · Tutor for 6 years
Answer
**4.****(a) Newtonian Momentum:*** p = mv* p = (1.67 x 10⁻²⁷ kg)(0.99c)* p = (1.67 x 10⁻²⁷ kg)(0.99 x 3 x 10⁸ m/s)* p ≈ 4.96 x 10⁻¹⁹ kg m/s**(b) Relativistic Momentum:*** p = γmv where γ = 1/√(1 - v²/c²)* First, calculate γ: γ = 1/√(1 - (0.99c)²/c²) = 1/√(1 - 0.9801) ≈ 7.09* p = (7.09)(1.67 x 10⁻²⁷ kg)(0.99c)* p ≈ 3.51 x 10⁻¹⁸ kg m/s**(c) Ratio of Relativistic to Newtonian Momentum:*** Ratio = (3.51 x 10⁻¹⁸ kg m/s) / (4.96 x 10⁻¹⁹ kg m/s) ≈ 7.08 (This is approximately equal to γ, as expected)**5.*** Given: p = 5mv* Relativistic momentum: p = γmv* Therefore, γ = 5* γ = 1/√(1 - v²/c²)* 5 = 1/√(1 - v²/c²)* Squaring both sides: 25 = 1/(1 - v²/c²)* 1 - v²/c² = 1/25* v²/c² = 1 - 1/25 = 24/25* v = c√(24/25)* v ≈ 0.98c**6.*** Electron momentum: pₑ = γₑmₑvₑ where γₑ = 1/√(1 - (0.999c)²/c²) ≈ 22.37* pₑ = (22.37)(9.11 x 10⁻³¹ kg)(0.999c) ≈ 6.26 x 10⁻²¹ kg m/s* Ship momentum: pₛ = mₛvₛ* Given pₑ = pₛ, so 6.26 x 10⁻²¹ kg m/s = (4.38 x 10⁷ kg)vₛ* vₛ ≈ 1.43 x 10⁻²⁸ m/s**7.**No, you would not feel compressed. Length contraction is relative. From your perspective within the spacecraft, you are at rest, and it is Earth that is moving at 0.99c. Therefore, you would observe Earth to be compressed in the direction of its motion, not yourself. Within your own frame of reference, your dimensions remain unchanged.**8.**We do not notice length contraction in everyday life because the speeds we typically encounter are much, much smaller than the speed of light. The Lorentz factor, γ, which determines the amount of length contraction, is very close to 1 at these speeds. The effects of length contraction become noticeable only at speeds approaching a significant fraction of the speed of light. Even for a car traveling at a high speed (e.g., 100 mph), the change in length due to relativistic effects is incredibly tiny and far too small to be perceived.