The left ventricle of the heart accelerates blood from rest to a velocity of +27 cm/s.
(a) If the displacement of the blood during the acceleration is +2.3 cm, determine its acceleration (in cm/s2).
(b) How much time does it take for the blood to reach its final velocity?
To solve this problem, we need to utilize the equations of motion. In this case, we can use the following equation:
v^2 = u^2 + 2as,
where:
v is the final velocity,
u is the initial velocity,
a is the acceleration,
s is the displacement.
(a) First, let's find the acceleration (a). We are given the final velocity (v) as +27 cm/s, the initial velocity (u) as 0 cm/s (since the blood starts from rest), and the displacement (s) as +2.3 cm. Plugging these values into the equation, we get:
(27 cm/s)^2 = (0 cm/s)^2 + 2a(2.3 cm),
729 cm^2/s^2 = 4.6a cm^2.
Now, we can solve for the acceleration (a):
a = 729 cm^2/s^2 / 4.6 cm^2,
a ≈ 158.7 cm/s^2.
Therefore, the acceleration of the blood is approximately 158.7 cm/s^2.
(b) To determine the time (t) it takes for the blood to reach its final velocity, we can use the following equation:
v = u + at,
where:
v is the final velocity,
u is the initial velocity,
a is the acceleration,
t is the time.
We already know the final velocity (v) as +27 cm/s, the initial velocity (u) as 0 cm/s, and the acceleration (a) as 158.7 cm/s^2. Plugging these values into the equation, we can solve for t:
27 cm/s = 0 cm/s + 158.7 cm/s^2 * t,
27 cm/s = 158.7 cm/s^2 * t.
Now, we can solve for t:
t = 27 cm/s / 158.7 cm/s^2,
t ≈ 0.17 s.
Therefore, it takes approximately 0.17 seconds for the blood to reach its final velocity.