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Unformatted text preview: ctor, s is the
moved distance by particle.
9 10 Engineering and Physics Examples Engineering and Physics Examples
Example: Velocity and Acceleration Example: Centripetal Acceleration The position of a moving particle is given by r
r = r0 cos ωtˆ + r0 sin ωtˆ
v = − r0ω sin ωtˆ + r0ω cos ωtˆ
a(t ) = r′′(t )
= −ω 2 (r0 cos ωtˆ + r0 sin ωtˆ)
= −ω r ,
i.e., is in the opposite
i.e., a is in the opposite dierction of r
of r (t ) = t 2 i + tj + 2.5tk
Graph the curve defined
Graph the curve defined by r (t ) and the vectors v (2) and a(2) .
Solution: x = t 2 , y = t , z = 2.5t
x = y 2 ⇒ Parabola
v (t ) = r′(t ) = 2ti + j + 2.5k
a(t ) = r′′(t ) = 2i with a =|| a ||, v =|| v || r (2) = 4i + 2 j + 5k ⇒ P(4,2,5)
v (2) = 4 i + j + 2.5k ∴ a = v 2 / r0
11 12 3 1/2/2011 Engineering and Physics Examples Example (Cont. ) Example: Curvilinear Motion in the Plane
A projectile is launched with an initial velocity
v 0 = v0 cos θ ˆ + v0 sin θ ˆ
and an initial height : s0 = s0 j v(t ) = (v0 cos θ)i + (− gt + v0 sin θ) j r
r (t ) = ∫ v (t )dt = (v0 cos θ )...
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- Winter '10