A particle moves at a constant speed along a curve on the surface of the sphere
ID: 2848313 • Letter: A
Question
A particle moves at a constant speed along a curve on the surface of the sphere x^2 + y^2+z^2 = 49 . Let r(t), v(t), a(t), and T(t) denote the particle's position, velocity, acceleration, and unit tangent vector at time t. Suppose these are all defined for all t. Which of the following must be true? (Write false, if false). Justify you answers.
a) For each t, ||a(t)|| = 0
b) For each t, a(t) . v(t) = 0
c) For each t, r(t) x a(t) = 0
d) For each t, r(t) and T(t) must be orthongonal
PLEASE EXPLAIN! Thanks :)
Explanation / Answer
1. false .
a(t) = Del^2 (x^2 + y^2+z^2 - 49) = 6
mode "a" = 6 for all t
b. true
a(t) = dV(t) / dt
a(t) will be perpendicular to V(t) so dot product will be zero .
a.V = av cos(90) = 0
c. true
a(t) = d^2 r(t) /dt
a(t) will be parallel to r(t).
axr = ar sin (0) = 0
d. true
r(t) is position vector
T(t) is tangential vector at r(t). so they will be perpendecular to each other.
r(t) and T(t) must be orthogonal.
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