To solve this problem, we can use the Lorentz transformation equations that relate the coordinates and time measurements of events observed in different reference frames moving relative to each other at constant velocity. Specifically, we will use the equation for the angle of an object as seen by an observer in a different frame of reference:
tan(θ') = (sin(θ) + v/c)/(γ(cos(θ) + v/c))
where θ is the angle of the object as measured in its own reference frame, θ' is the angle as measured in the observer's frame, v is the relative velocity between the two frames, c is the speed of light, and γ is the Lorentz factor, given by:
γ = 1/sqrt(1 - v^2/c^2)
In this problem, Jasmine is in a spaceship traveling at 0.84 c, and she holds a phone at a 21 ∘ angle to take a photo of the surroundings. We want to find the angle that the phone makes with the direction of travel as seen by Heather standing still on Earth.
First, we need to calculate the Lorentz factor:
γ = 1/sqrt(1 - v^2/c^2) = 1/sqrt(1 - 0.84^2) ≈ 1.94
Next, we can use the equation for the angle as seen by the observer in the other reference frame:
tan(θ') = (sin(θ) + v/c)/(γ(cos(θ) + v/c))
In this case, θ = 21 ∘, v = 0.84 c, and γ = 1.94. Plugging in these values, we get:
tan(θ') = (sin(21 ∘) + 0.84 c/c)/(1.94(cos(21 ∘) + 0.84 c/c))
tan(θ') = (0.358 + 0.84)/(1.94(0.927 + 0.84))
tan(θ') ≈ 0.710
Finally, we can take the inverse tangent of both sides to solve for θ':
θ' = tan^-1(0.710) ≈ 34.7 ∘
Therefore, Heather on Earth sees the phone at an angle of approximately 34.7 ∘ with the direction of travel of the spaceship.