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A cannon is placed on the surface of the Earth at colatitude (polar angle) and pointed due east. The cannon barrel makes an angle a with the horizontal, and the projectile emerges with initial speed vo. The rotational speed of the Earth around its polar axis is w. (Hints: use the expansion F(t) = To(t) + F(t) in powers of w; use a coordinate system with a pointing South along the surface, y pointing East along the surface, and 2 pointing up perpendicularly

A cannon is placed on the surface of the Earth at colatitude (polar angle) and pointed due east. The cannon barrel makes an angle a with the horizontal, and the projectile emerges with initial speed vo. The rotational speed of the Earth around its polar axis is w. (Hints: use the expansion F(t) = To(t) + F(t) in powers of w; use a coordinate system with a pointing South along the surface, y pointing East along the surface, and 2 pointing up perpendicularly

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3 of 3 QD 5. A cannon is placed on the surface of the Earth at colatitude (polar angle) 0 and pointed due east. The cannon barrel makes an angle a with the horizontal, and the projectile emerges with initial speed vo. The rotational speed of the Earth around its polar axis is w. (Hints: use the expansion r(t) = To(t) + ri(t) in powers of w; use a coordinate system with a pointing South along the surface, ŷ pointing East along the surface, and 2 pointing up perpendicularly 1 USD/EUR +0.37% ESC Read aloud to the surface; write was a linear combination of the â,ŷ and 2 unit vectors; and assume that the horizontal distance traversed by the projectile is so small compared to the Earth's radius that the Earth's surface can be considered flat.) (a) Compute, to first order in w, and in terms of vo, g, w, 0, and a, the lateral deflection of a projectile when it strikes the Earth. Give magnitude and direction (in terms of N, S, E, W) for the deflection. (b) If R is the range of the projectile for the case w=0, compute, to first order in w, and in terms of R, g, w, 0, and tan(a), the change in the range. FE Fa 7 ‒‒ Q Search HH FH FS FE P F7 G FB FB 0 FI → F11 + 94x6 BACKSPACE 2:57 PM 12/5/2023 O O A
Transcribed Image Text:3 of 3 QD 5. A cannon is placed on the surface of the Earth at colatitude (polar angle) 0 and pointed due east. The cannon barrel makes an angle a with the horizontal, and the projectile emerges with initial speed vo. The rotational speed of the Earth around its polar axis is w. (Hints: use the expansion r(t) = To(t) + ri(t) in powers of w; use a coordinate system with a pointing South along the surface, ŷ pointing East along the surface, and 2 pointing up perpendicularly 1 USD/EUR +0.37% ESC Read aloud to the surface; write was a linear combination of the â,ŷ and 2 unit vectors; and assume that the horizontal distance traversed by the projectile is so small compared to the Earth's radius that the Earth's surface can be considered flat.) (a) Compute, to first order in w, and in terms of vo, g, w, 0, and a, the lateral deflection of a projectile when it strikes the Earth. Give magnitude and direction (in terms of N, S, E, W) for the deflection. (b) If R is the range of the projectile for the case w=0, compute, to first order in w, and in terms of R, g, w, 0, and tan(a), the change in the range. FE Fa 7 ‒‒ Q Search HH FH FS FE P F7 G FB FB 0 FI → F11 + 94x6 BACKSPACE 2:57 PM 12/5/2023 O O A
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