Tina wants to install one unit that both heats and cools. Which should Tina install?

An alien spaceship streaks past Spanos football stadium along the direction of play at 0.8c. A football field is 120 yards long

Viktor [21]

Answer:

(i) Length of stadium, L₀ = 72 yards; width of stadium W = 55 yards (ii) 5 hours (iii) The people on earth. This is because they are not moving at relativistic speed relative to the game. (iv) The people on earth. This is because they are not moving at relativistic speed relative to the game.

Explanation:

(i) Given Length of stadium, L₀ = 120 yards, width of stadium, W₀ = 55 yards, speed of alien spaceship in direction of play, v = 0.8c

We use the equation for length contraction to determine the length of stadium L measured by the alien spaceship, since it is moving in the direction of game play and length contraction only occurs in the direction of motion. So, L = $\sqrt{1-\beta^{2} }$ L₀ where β = $\frac{v}{c}$ v= speed of alien spaceship and c= speed of light. So, β = $\frac{v}{c}$ = $\frac{0.8c}{c}$ = 0.8. Therefore, L = $\sqrt{1-0.8^{2} }$ L₀= $\sqrt{1-0.64}$ L₀= $\sqrt{0.36}$ L₀ = 0.6L₀ = 0.6 × 120 = 72 yards. So, the alien spaceship measures a length of 72 yards. The alien spaceship measures a width,W of 55 yards because there is no length contraction in the direction perpendicular to that of its motion. So, W = W₀ = 55 yards.

(ii) Since the game begins at 1:30 PM Pacific Standard Time (PST) and ends 4:30 PM Pacific Standard Time (PST), the proper time t₀, which is the duration of the event is 4:30 - 1:30 = 3 hours. The time measured by the alien spaceship t (since the time dilates)is given by t = t₀/ $\sqrt{1 - \beta ^{2} }$ . From (i) above β = 0.8 and t₀ = 3 hours. So, t = 3/ $\sqrt{1 - 0.8^{2} }$ = 3/0.6 = 5 hours.

(iii) The people in Spanos football stadium. This is because they are not moving at relativistic speed relative to the game. Since $\frac{v}{c}$ ≈ 0 then β ≈ 0 So, L = $\sqrt{1-\beta^{2} }$ L₀= So, L = $\sqrt{1-0}$ L₀thus L = L₀ for the spectators in spanos football field (iv) The people in Spanos football stadium. This is because they are not moving at relativistic speed relative to the game. Since $\frac{v}{c}$ ≈ 0 then β ≈ 0. So, t = t₀/ $\sqrt{1 - \beta ^{2} }$ = t₀/ $\sqrt{1-0}$ = t₀.Thus t = t₀ for the spectators in spanos football field.