Before starting this problem, review Multiple-Concept Example 6. The left ventricle of the heart accelerates blood from rest to a velocity of +20.6 cm/s. (a) If the displacement of the blood during the acceleration is +2.87 cm, determine its acceleration (in cm/s2). (b) How much time does blood take to reach its final velocity?
To find the answers to these questions, we need to use the equations of motion for uniformly accelerated linear motion. Let's go step by step.
(a) To determine the acceleration first we need to identify the known values. We are given:
Initial velocity (u) = 0 cm/s (since the blood starts from rest)
Final velocity (v) = +20.6 cm/s
Displacement (s) = +2.87 cm
We can use the equation:
v^2 = u^2 + 2as
where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the displacement.
Rearranging the equation to solve for acceleration (a), we have:
a = (v^2 - u^2) / (2s)
Substituting the known values:
a = (+20.6 cm/s)^2 - (0 cm/s)^2 / (2 * +2.87 cm)
Simplifying further:
a = 424.36 cm^2/s^2 / (5.74 cm)
Finally, we calculate the acceleration:
a ≈ 74 cm/s^2
Therefore, the acceleration of the blood is approximately +74 cm/s^2.
(b) To find the time it takes for the blood to reach its final velocity, we can use the equation:
v = u + at
where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time taken.
Rearranging the equation to solve for time (t):
t = (v - u) / a
Substituting the known values:
t = (+20.6 cm/s - 0 cm/s) / (+74 cm/s^2)
Simplifying further:
t = +20.6 cm/s / +74 cm/s^2
Finally, we calculate the time:
t ≈ 0.278 seconds
Therefore, it takes approximately 0.278 seconds for the blood to reach its final velocity.
To solve the problem, we used the equations of motion for uniformly accelerated linear motion, specifically the equations v^2 = u^2 + 2as and v = u + at.