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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 Spring '08
 CHAN
 mechanics, Derivative, 3 pts, 2 pts, 20 min

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