Problem 69 - University Physics with Modern Physics

Question

Bl0 Whiplash Injuries. When a car is hit from behind, its passengers undergo sudden forward acceleration, which can cause a severe neck injury known as whiplash. During normal acceleration, the neck muscles play a large role in accelerating the head so that the bones are not injured. But during a very sudden acceleration, the muscles do not react immediately because they are flexible, so most of the accelerating force is provided by the neck bones. Experimental tests have shown that these bones will fracture if they absorb more than 8.0 \(\mathrm{J}\) of energy. (a) If a car waiting at a stoplight is rear-ended in a collision that lasts for 10.0 \(\mathrm{ms},\) what is the greatest speed this car and its driver can reach without breaking neck bones if the driver's head has a mass of 5.0 \(\mathrm{kg}\) (which is about right for a 70 -kg person)? Express your answer in \(\mathrm{m} / \mathrm{s}\) and in mph. (b) What is the acceleration of the passengers during the collision in part (a), and how large a force is acting to accelerate their heads? Express the acceleration in \(\mathrm{m} / \mathrm{s}^{2}\) and in \(g^{\prime} \mathrm{s}\) .

Step-by-Step Solution

Answer

Greatest speed: 1.79 m/s or 4.01 mph; Acceleration: 179 m/s² or 18.24 g's; Force: 895 N.

Step 1: Understanding the Problem

We have to determine the greatest speed a driver's head can reach during a whiplash injury without the neck bones fracturing when the car is rear-ended. The head's mass is 5 kg and it must absorb less than 8 J to avoid injury. Additionally, we need to compute the acceleration and force during the collision.

Step 2: Kinetic Energy Calculation

The energy absorbed by the neck bones can be modeled as kinetic energy: \( E = \frac{1}{2} m v^2 \).Here, \( E = 8 \text{ J} \) and \( m = 5 \text{ kg} \). We solve for \( v \), the greatest speed:\[ 8 = \frac{1}{2} \times 5 \times v^2. \]Solving this gives \( v = \sqrt{\frac{16}{5}} = \sqrt{3.2} \approx 1.79 \text{ m/s}. \)

Step 3: Convert Speed to Miles per Hour

Convert the greatest speed from m/s to mph using the conversion factor (1 m/s = 2.237 mph):\[ 1.79 \text{ m/s} \times 2.237 = 4.01 \text{ mph}. \]

Step 4: Time-Dependent Acceleration Calculation

The time for the collision is 10 ms, or 0.01 s. We use \( v = at \) to find the acceleration \( a \):\[ a = \frac{v}{t} = \frac{1.79}{0.01} = 179 \text{ m/s}^2. \]

Step 5: Convert Acceleration to G's

Convert the acceleration to g's (where 1 g = 9.81 m/s²):\[ \frac{179}{9.81} \approx 18.24 \text{ g's}. \]

Step 6: Force Calculation

Determine the force using \( F = ma \):\[ F = 5 \times 179 = 895 \text{ N}. \]

Key Concepts

Kinetic Energy

In the context of whiplash injuries from rear-end collisions, kinetic energy plays a vital role in determining whether neck bones will fracture. Kinetic energy is the energy that an object possesses due to its motion and is given by the formula:
\[ E = \frac{1}{2} m v^2 \]

• \( E \): Kinetic energy in joules (J)
• \( m \): Mass of the object in kilograms (kg) – for this scenario, the head's mass is 5 kg
• \( v \): Velocity in meters per second (m/s)

Understanding how kinetic energy works helps in assessing the potential for bone fractures. If the neck bones absorb more than 8 J of energy, fractures could occur. By substituting the values and solving, we find that the greatest speed the car can reach without causing injury to the driver's neck is approximately 1.79 m/s. This indicates how energy constraints govern safety thresholds in collisions.
Converting speeds and analyzing such scenarios provide valuable insights into preventive measures for reducing whiplash injuries.

Acceleration Calculation

Acceleration is a crucial concept when analyzing the sudden movements experienced during a collision. It involves changes in velocity over time and can significantly impact the force exerted on passengers.
To find the acceleration experienced by passengers in a collision lasting 10 milliseconds (0.01 seconds), the formula\( v = at \)is used, rearranging to: \[ a = \frac{v}{t} \]

• \( v \): Final velocity, which is 1.79 m/s from previous calculations
• \( t \): Time in seconds (0.01 s)

Plugging in these values, the acceleration is determined to be 179 m/s². This rapid acceleration, expressed in m/s², demonstrates the drastic change in motion that occurs in mere milliseconds during an impact.
Such high acceleration values explain why severe whiplash injuries can occur even within short-duration collisions, emphasizing the need for safety technologies designed to mitigate these effects.

Force Calculation

Force, when studied in the context of a collision, provides an understanding of the strength exerted on bodies during impact. It is crucial for evaluating the potential for injuries such as whiplash.
The force that acts on an object is calculated with Newton’s second law of motion:
\[ F = ma\]

• \( F \): Force in newtons (N)
• \( m \): Mass of the object, specifically the head, which is 5 kg
• \( a \): Acceleration, previously calculated as 179 m/s²

By substituting the given values, the force acting on the driver’s head is found to be 895 N. This significant force helps explain the potential for injury, as it places tremendous stress on the neck muscles and bones.
Understanding force calculation helps in the design of safer vehicles and protective gear, potentially reducing the severity of whiplash injuries during collisions.

Collision Physics

Collision physics revolves around understanding the interactions during impact and how they affect objects involved in the collision. When a vehicle is rear-ended, the sudden transfer of forces and energy can inflict severe damage unless mitigated. Instances of whiplash highlight the physics of collisions on a very personal level. The rear impact causes the car – and by extension, the occupant’s head – to accelerate forward quickly. This interaction involves:

• Transfer of kinetic energy from the colliding vehicle to the one at rest
• Rapid acceleration of the vehicle and passenger's body
• Adverse forces acting on sensitive areas like the neck

Physics principles help explain why certain safety standards and technologies, like headrests and seatbelts, are crucial. They absorb some impact forces, distribute them more evenly, and reduce acceleration rates.
Understanding collision physics aids not only in recognizing probable injury mechanisms but also in engineering improvements in vehicle safety to protect occupants.