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Chapter 2: Problem 27

Before starting this problem, review Multiple-Concept Example \(6 .\) The left ventricle of the heart accelerates blood from rest to a velocity of \(+26 \mathrm{cm} / \mathrm{s}\) . (a) If the displacement of the blood during the acceleration is \(+2.0 \mathrm{cm},\) determine its acceleration (in \(\mathrm{cm} / \mathrm{s}^{2} )\) (b) How much time does blood take to reach its final velocity?

Short Answer

Expert verified
(a) Acceleration is 169 cm/s虏. (b) Time taken is approximately 0.154 seconds.

Step by step solution

01

Identify Known Values

The problem provides us with the following information:- Initial velocity \( v_i = 0 \) cm/s (since the blood starts from rest)- Final velocity \( v_f = 26 \) cm/s- Displacement \( s = 2.0 \) cm. We'll use these values to find acceleration using the kinematic equation.
02

Use Kinematic Equation to Find Acceleration

We can use the kinematic equation:\[ v_f^2 = v_i^2 + 2a s \]Substituting the known values:\[ (26)^2 = 0 + 2a(2.0) \]This simplifies to:\[ 676 = 4a \]From which we can solve for the acceleration \( a \):\[ a = \frac{676}{4} = 169 \text{ cm/s}^2 \]
03

Use Another Kinematic Equation to Find Time

Now, we need to find the time taken using the equation:\[ v_f = v_i + at \]Substitute the known values for \( v_f, v_i, \) and \( a \) to solve for \( t \):\[ 26 = 0 + 169t \]\[ t = \frac{26}{169} \approx 0.154 \text{ seconds} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Acceleration
Acceleration is a measure of how quickly the velocity of an object changes over time. In kinematics, it is an essential concept that helps explain motion dynamics. When an object speeds up, slows down, or changes direction, it experiences acceleration. The formula for calculating acceleration is:\[a = \frac{\Delta v}{\Delta t}\],where \( \Delta v \) is the change in velocity and \( \Delta t \) is the time over which this change occurs.
  • In the heart problem, we initial velocity \( v_i = 0 \text{ cm/s} \) and final velocity \( v_f = 26 \text{ cm/s}\) to calculate acceleration.
  • The displacement \( s = 2 \text{ cm} \) during this change helps apply the proper kinematic equation.
Understanding acceleration in the context of the heart exercise shows how blood speeds up from rest to reach a certain velocity.
Kinematic Equations
Kinematic equations are a set of formulas that describe the motion of objects under uniform acceleration. These equations are crucial in solving problems involving acceleration, velocity, displacement, and time. They relate four primary variables of motion, allowing us to solve for any one of them if the others are known.
  • The equation used in the example is \( v_f^2 = v_i^2 + 2as \),which connects final and initial velocity, acceleration, and displacement.
  • Another helpful kinematic equation is \( v_f = v_i + at \), used to solve for time in this problem.
Using these equations accurately is key to analyzing motion effectively. Plugging in the known values, like initial velocity and displacement, allows for solving the desired unknowns like acceleration and time.
Velocity
Velocity refers to the speed of an object in a particular direction. It is a vector, meaning it has both magnitude and direction, unlike speed which only has magnitude. In the given exercise, understanding velocity is crucial because it starts from zero and reaches a known final value.
  • Initial velocity \( v_i = 0 \) implies the object starts from rest, making calculations simpler.
  • Final velocity \( v_f = 26 \text{ cm/s}\) tells us how fast the blood is moving at the end of acceleration.
By understanding changes in velocity, we can further explore the nature of the object's motion. The heart's left ventricle demonstrates the practical application of velocity/velocity change in biological systems.
Displacement
Displacement is a vector quantity that refers to the overall change in position of an object. It is different from distance, as it considers the straight-line path between the start and end points.
  • In this exercise, the displacement \( s = 2 \text{ cm} \) is directly tied to how far the blood has moved while accelerating.
  • Knowing displacement helps apply the kinematic equations accurately.
In problems like these, displacement provides a way to quantify and analyze movement from point A to point B. It shows how far and in what direction an object has moved, not just how fast.

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Most popular questions from this chapter

A ball is thrown vertically upward, which is the positive direction. A little later it returns to its point of release. The ball is in the air for a total time of 8.0 s. What is its initial velocity? Neglect air resistance.

A golf ball is dropped from rest from a height of 9.50 \(\mathrm{m} .\) It hits the pavement, then bounces back up, rising just 5.70 \(\mathrm{m}\) before falling back down again. A boy then catches the ball on the way down when it is 1.20 \(\mathrm{m}\) above the pavement. Ignoring air resistance, calculate the total amount of time that the ball is in the air, from drop to catch.

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An Australian emu is running due north in a straight line at a speed of 13.0 \(\mathrm{m} / \mathrm{s}\) and slows down to a speed of 10.6 \(\mathrm{m} / \mathrm{s}\) in 4.0 s. (a) What is the direction of the bird's acceleration? (b) Assuming that the acceleration remain the same, what is the bird's velocity after an additional 2.0 \(\mathrm{s}\) has elapsed?

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