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Problem Set 2 Solutions

# Problem Set 2 Solutions - 1 Physics 200a PSII 1 Let A = 3i...

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1 Physics 200a PSII 1. Let A = 3 i + 4 j and B = 5 i - 6 j . (i) Find A + B , A - B , 2 A + 3 B , and C such that A + B + C = 0 . (ii) Find A , the length of A and the angle it makes with the x-axis. A + B = 8 i - 2 j A - B = - 2 i + 10 j 2 A + 3 B = 6 i + 8 j + 15 i - 18 j = 21 i - 10 j C = - A - B = - 8 i + 2 j ( ii ) A = 3 2 + 4 2 = 5 2. A train is moving with velocity v TG = 3 i + 4 j relative to the ground. A bullet is ﬁred in the train with velocity v BT = 15 i - 6 j relative to the train. What is the bullets’ velocity v BG relative to the ground? v BG = v BT + v = 18 i - 2 j 3. Consider the primed axis rotated relative to the unprimed by an angle φ in the coun- terclockwise direction. (i) Derive the relation A x = A 0 x cos φ - A 0 y sin φ A y = A 0 y cos φ + A 0 x sin φ that expresses unprimed components in terms of primed components of a vector ~ A using class notes if needed to get started. First note (by drawing a ﬁgure) that the rotated unit vectors are related to the old ones as follows i 0 = i cos φ + j sin φ j 0 = j cos φ - i sin φ Now we have A = A 0 x i 0 + A 0 y j 0 (1) = A 0 x ( i cos φ + j sin φ ) + A 0 y ( j cos φ - i sin φ ) (2) = i ( A 0 x cos φ - A 0 y sin φ ) + j ( A 0 y cos φ + A 0 x sin φ ) (3) The coeﬃcients of i and j , are by deﬁnition A x and A y , yielding the desired result. (ii) Invert these relations to express the primed components in terms of unprimed com- ponents. In doing this remember that the sines and cosines are constants and that we should treat A 0 x and A 0 y as unknowns written in terms of knowns A x and A y . (Thus multiply one equation by something, another by something else, add and subtract etc to isolate the un- knowns. Use simple trig identities)

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2 To ﬁnd A 0 x , we can multiply the equation for A x by cos φ , the equation for A y by sin φ to get A x cos φ = A 0 x cos 2 φ - A 0 y sin φ cos φ (4) A y sin φ = A 0 y sin φ cos φ + A 0 x sin 2 φ. (5) Adding the two equations gives A x cos φ + A y sin φ = A 0 x (cos 2 φ + sin 2 φ ) = A 0 x , because the terms containing A 0 y cancel. Similarly, to ﬁnd
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Problem Set 2 Solutions - 1 Physics 200a PSII 1 Let A = 3i...

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