86. Medical, Calc The velocity of the wall of a beating heart is given by the fo

Question

86. Medical, Calc The velocity of the wall of a beating heart is given by the following function: v(t) = 0.023sin(7t) (SI units) (a) Calculate the maximum speed of the heart during one contraction. (b) Write an expression for the position of the heart as a function of time (x(t) = ?). Explain what the various terms in the expression mean physically. Assume that the heart is at its maximum displacement from equilibrium when t = 0. (c) Determine the total distance (not displacement) that the heart moves in 45 s.

Explanation / Answer

a).
for maximum speed,
dv/dt = 0
=> 0.023 *cos(7*t) *7 = 0
=> cos(7t) = 0
=> 7t = pi/2
=> t = 0.224 s

v = 0.023*sin(pi/2) = 0.023 m/s

b).
v = dx/dt
=> x(t) = integration of v(t) w.r.t. 't'
=> x(t) = [-0.023*cos(7*t)]/7 + c

where, c = constant of integration

when t=0, x = maximum
so, c = 0

=> x(t) = [-0.023*cos(7*t)]/7

c).
x = [-0.023*cos(7*45)]/7 = 2.32*10^-3 m = 2.32 mm

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