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Chapter 2: Problem 26

\(\textbf{Prevention of Hip Fractures.}\) Falls resulting in hip fractures are a major cause of injury and even death to the elderly. Typically, the hip鈥檚 speed at impact is about 2.0 m/s. If this can be reduced to 1.3 m/s or less, the hip will usually not fracture. One way to do this is by wearing elastic hip pads. (a) If a typical pad is 5.0 cm thick and compresses by 2.0 cm during the impact of a fall, what constant acceleration (in m/s\(^{2}\) and in \(\text{g}\)鈥檚) does the hip undergo to reduce its speed from 2.0 m/s to 1.3 m/s? (b) The acceleration you found in part (a) may seem rather large, but to assess its effects on the hip, calculate how long it lasts.

Short Answer

Expert verified
The constant acceleration is -57.75 m/s\(^{2}\) (about -5.89 g's) and lasts about 0.012 seconds.

Step by step solution

01

Identify known values

We know: Initial velocity \( v_i = 2.0 \, \text{m/s} \), Final velocity \( v_f = 1.3 \, \text{m/s} \), Compression distance \( d = 0.02 \, \text{m} \) (compresses by 2.0 cm).
02

Use kinematic equation

The kinematic equation \( v_f^2 = v_i^2 + 2a d \) can be used to find the acceleration \( a \). Substitute the known values into the equation: \( (1.3)^2 = (2.0)^2 + 2a(0.02) \).
03

Solve for acceleration

Rearrange the equation to \( a = \frac{(1.3)^2 - (2.0)^2}{2 \times 0.02} \). Calculate: \( a = \frac{1.69 - 4}{0.04} = \frac{-2.31}{0.04} = -57.75 \, \text{m/s}^2 \).
04

Convert acceleration to g's

To convert \( a \) from m/s\(^2\) to g's, divide by \( g = 9.81 \, \text{m/s}^2 \). Thus, \( a = \frac{-57.75}{9.81} \approx -5.89 \, \text{g} \).
05

Calculate time duration of acceleration

Use the equation \( v_f = v_i + at \) to solve for \( t \). Rearrange the equation to \( t = \frac{v_f - v_i}{a} \). Substituting in values: \( t = \frac{1.3 - 2.0}{-57.75} \approx 0.012 \, \text{seconds} \).

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Key Concepts

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

Understanding Constant Acceleration
Constant acceleration occurs when the rate of change of velocity remains unchanged over time. This means the acceleration is uniform throughout the motion. In our problem, we're focusing on the moment when a hip pad slows down the fall of a person to prevent a fracture.
In the scenario of the falling hip, compression of the pad results in a constant acceleration that changes the velocity from 2.0 m/s to 1.3 m/s. We calculate this by assuming the acceleration does not waver while the pad compresses. This understanding is crucial when analyzing how effective pads can be in reducing impact forces during a fall.
  • Constant acceleration can simplify complex motions.
  • It helps in designing safety measures like hip pads.
  • Allows use of straightforward equations to model real-world events.
Recognizing constant acceleration's role in the scenario helps us understand the padding's function. It converts a sudden stop into a slightly protracted stop, lowering impact velocity and, thereby, injury risk.
The Role of Kinematic Equations
Kinematic equations are mathematical formulas used to describe motion in terms of displacement, velocity, acceleration, and time. In this exercise, they help us figure out how much acceleration the pad provides.
We specifically use the kinematic equation: \[ v_f^2 = v_i^2 + 2a d \],where- \( v_f \) is the final velocity,- \( v_i \) is the initial velocity,- \( a \) is the acceleration, and- \( d \) is the displacement.
This formula derives acceleration when we have changes in speed over a known distance. It's very handy in cases where constant acceleration is assumed, like the cushioning effect of hip pads.
  • They enable precise calculations using initial conditions and some known values.
  • Allows us to deduce time, velocity, or acceleration involved in a motion.
  • Helps predict outcomes of safety measures like hip pads before physical testing.
In essence, kinematic equations bridge theoretical physics and practical safety engineering, supporting design decisions that safeguard health.
Importance of Hip Fracture Prevention
Preventing hip fractures is vital, especially among the elderly, due to risks like hospitalization and decreased mobility.
When a person falls, the force at which their hip hits the ground can mean the difference between a bruise and a fracture. By reducing the impact speed with mechanisms like hip pads, the likelihood of injury reduces significantly.
  • Allows elderly to maintain independence longer.
  • Decreases healthcare costs associated with injuries.
  • Improves life quality by reducing recovery times.
Hip fracture prevention methods like padded clothing or hip pads are simple yet powerful. They employ basic kinematic principles to diffuse potentially harmful forces by slowing down the body's impact speed, offering a practical and effective safety solution.

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Most popular questions from this chapter

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