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Prevention of Hip Fractures. Falls resulting in hip fractures are a major cause of injury and even death to the elderly. Typically, the hip's speed at impact is about 2.0 \(\mathrm{m} / \mathrm{s} .\) If this can be reduced to 1.3 \(\mathrm{m} / \mathrm{s}\) 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 \(\mathrm{cm}\) thick and compresses by 2.0 \(\mathrm{cm}\) during the impact of a fall, what constant acceleration \(\operatorname{m} / \mathrm{s}^{2}\) and in \(g^{\prime} \mathrm{s}\) ) does the hip undergo to reduce its speed from 2.0 \(\mathrm{m} / \mathrm{s}\) to 1.3 \(\mathrm{m} / \mathrm{s} ?\) (b) The acceleration you found in part (a) may seem rather large, but to fully assess its effects on the hip, calculate how long it lasts.

Step by step solution

01

Identify Relevant Equations

To start, identify the relevant kinematic equation that relates initial velocity (\(v_i\)), final velocity (\(v_f\)), acceleration (\(a\)), and displacement (\(d\)):\[v_f^2 = v_i^2 + 2ad\]

02

Assign Known Values

Assign values: initial speed \(v_i = 2.0 \, \mathrm{m/s}\), final speed \(v_f = 1.3 \, \mathrm{m/s}\), and compression \(d = 2.0 \, \mathrm{cm} = 0.020 \, \mathrm{m}\).

03

Rearrange for Acceleration

Rearrange the equation to solve for acceleration \(a\):\[a = \frac{v_f^2 - v_i^2}{2d}\]

04

Substitute and Solve for Acceleration

Substitute the known values into the equation:\[a = \frac{(1.3)^2 - (2.0)^2}{2 \times 0.020}\]Calculate \(a\):\[a = \frac{1.69 - 4.00}{0.040} = \frac{-2.31}{0.040} = -57.75 \, \mathrm{m/s^2}\]

05

Convert Acceleration to g's

Convert the acceleration to gravitational units (where \(1 \, g = 9.81 \, \mathrm{m/s^2}\)):\[a = \frac{-57.75}{9.81} \approx -5.89 \, g\]

06

Calculate Duration of Acceleration

Use a kinematic equation to find the duration \(t\) of the acceleration:\[v_f = v_i + at\]Rearrange to solve for \(t\):\[t = \frac{v_f - v_i}{a}\]Substitute the known values:\[t = \frac{1.3 - 2.0}{-57.75} = \frac{-0.7}{-57.75} \approx 0.012 \, \mathrm{s}\]

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Key Concepts

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

Acceleration Calculation

When calculating acceleration, it's essential to determine how quickly the speed of an object changes over time. In this case, we are examining a scenario where the impact speed of a hip during a fall needs to be reduced to prevent injury.
To calculate the acceleration, we use the kinematic equation that connects initial velocity (\(v_i\)), final velocity (\(v_f\)), acceleration (\(a\)), and displacement (\(d\)). This equation is given by:\[v_f^2 = v_i^2 + 2ad\]

• Initial velocity (\(v_i = 2.0 \;\mathrm{m/s}\)) is the speed before impact.
• Final velocity (\(v_f = 1.3 \;\mathrm{m/s}\)) is the desired speed post-impact.
• Displacement (\(d = 0.020 \;\mathrm{m}\)) is the compression distance of the pad.

Rearranging the equation to solve for acceleration, we find:\[a = \frac{v_f^2 - v_i^2}{2d}\]Inserting the known values, the formula becomes:\[a = \frac{(1.3)^2 - (2.0)^2}{2 \times 0.020}\] The calculated acceleration is \(a = -57.75 \;\mathrm{m/s^2}\), where the negative sign indicates a decrease in speed.

Impact Velocity Reduction

Reducing the velocity at which a hip hits the ground during a fall is crucial for minimizing injury risk. The goal is to bring the speed down from \(2.0 \;\mathrm{m/s}\) to \(1.3 \;\mathrm{m/s}\) or less. This reduction is achieved by compressing an elastic hip pad.
When the pad is compressed, it absorbs some of the fall's energy, decreasing the velocity of the impact.
The compression distance in this scenario is \(2.0 \;\mathrm{cm} = 0.020 \;\mathrm{m}\). This short distance is critical as it determines the efficiency of the hip pad in absorbing energy.
Key elements for impact velocity reduction include:

• Adequate padding thickness to absorb energy effectively.
• Materials in the pad that compress quickly upon impact and return to their original shape.
• Proper alignment so the hip pad takes the full brunt of the fall.

These factors work together to ensure that the speed is reduced effectively, minimizing the chance of fracture.

Kinematic Equations

Kinematic equations are powerful tools used in physics to describe the motion of objects. They are especially useful when dealing with situations that involve constant acceleration, like the case of a falling object protected by a pad.
The kinematic equation used here is: \[v_f^2 = v_i^2 + 2ad\]This equation relates:

• Initial velocity (\(v_i\))
• Final velocity (\(v_f\))
• Acceleration (\(a\))
• Displacement (\(d\))

The equation can be rearranged to solve for any of the unknowns, such as acceleration or time (\(t\)).
In this exercise, we also used another kinematic equation to calculate time:\[v_f = v_i + at\]Rearranged for time:\[t = \frac{v_f - v_i}{a}\]By plugging in the known values, we find the duration of the acceleration to be approximately \(0.012 \;\mathrm{s}\). This short time indicates how quickly the protective measures must act to reduce the impact velocity.