~ just means vector
Suppose ~r(u) = r0(u)~i + r1(u)~j satisfies r'0(u) > 0 and r'1(u) < 0. Let a particle begin traveling from the point ~r(0) along the path ~r(u) beginning at a standstill. A consequence p of the conservation of energy is that the speed of the particle when it reaches point ~r(u) is sqrt(2g(r1(0) - r1(u)) where g is acceleration due to gravity.
(a) From this fact, derive a formula for the time it takes a particle to descend from ~r(0) to ~r(u) for any fixed u.
(b) Suppose that ~r(t) = (t-sin(t))~i+ (1+ cos(t))~j. Compute the time to descend from ~r(u0) to ~r(pi) as well as the arc length of the path for u0 = 0, 0.5, 1. You may use a computer algebra system for the latter integrals, the first should be done by hand.
(c) Compute time of descent and arc length of at least two other paths connecting (0, 2) and (pi, 0). You may use a computer algebra system to evaluate the integrals.
(d) Make some conjectures about the path ~r(t) = (t - sin(t))~i + (1 + cos(t))~j, state evidence supporting and also explain how conclusive or inconclusive the evidence is.
~ just means vector
Suppose ~r(u) = r0(u)~i + r1(u)~j satisfies r'0(u) > 0 and r'1(u) < 0. Let a particle begin traveling from the point ~r(0) along the path ~r(u) beginning at a standstill. A consequence p of the conservation of energy is that the speed of the particle when it reaches point ~r(u) is sqrt(2g(r1(0) - r1(u)) where g is acceleration due to gravity.
(a) From this fact, derive a formula for the time it takes a particle to descend from ~r(0) to ~r(u) for any fixed u.
(b) Suppose that ~r(t) = (t-sin(t))~i+ (1+ cos(t))~j. Compute the time to descend from ~r(u0) to ~r(pi) as well as the arc length of the path for u0 = 0, 0.5, 1. You may use a computer algebra system for the latter integrals, the first should be done by hand.
(c) Compute time of descent and arc length of at least two other paths connecting (0, 2) and (pi, 0). You may use a computer algebra system to evaluate the integrals.
(d) Make some conjectures about the path ~r(t) = (t - sin(t))~i + (1 + cos(t))~j, state evidence supporting and also explain how conclusive or inconclusive the evidence is.
