# q01 - Physics 3221 Mechanics I Fall Term 2010 Quiz 1 This...

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Unformatted text preview: Physics 3221 Mechanics I Fall Term 2010 Quiz 1 This is a 20 min. quiz (closed book). There are two problems (the second problem is on the back). Problem 1. [2 pts] Find the gradients of the two functions u(x, y ) = x + y2 , x v (x, y ) = y + and show that they are always orthogonal. Solution. ∇u = 1 − y y2 e1 + 2 e2 2 x x x x2 ∇v = 2 e1 + 1 − 2 e2 y y ∇u · ∇v = 0 ⇒ 1 ∇u ⊥ ∇v x2 , y Problem 2. [3 pts.] The position vector of a point particle is given by r(t) = (3t2 − 4) e1 + t3 e2 + (t + 3) e3 . (a) ﬁnd the velocity v (t), the acceleration a(t) and the speed v (t) of this particle. (b) ﬁnd the time derivative of the vector r × v, i.e. ﬁnd d (r × v ) dt Solution. (a) v(t) = 6te1 + 3t2 e2 + e3 a(t) = 6e1 + 6te2 |v(t)| = √ 36t2 + 9t4 + 1 (b) d (r × v ) = v × v + r × a = r × a = −6t(t + 3)e1 + 6(t + 3)e2 + 12t(t2 − 2)e3 dt 2 ...
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