Question

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; most of the accelerating force is provided by the neck bones. Experiments have shown that these bones will fracture if they absorb more than 8.0 J of energy. (a) If a car waiting at a stoplight is rear-ended in a collision that lasts for 10.0 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 kg (which is about right for a 70-kg person)? Express your answer in m/s and in mi/h. (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 m/s\(^2\) and in \(g\)'s.

Short answer

(a) Max speed is 1.79 m/s (4.01 mi/h). (b) Acceleration is 179 m/s² (18.24 g), force is 895 N.

Step-by-step solution

Understand the problem

We're tasked to find the greatest speed the head can reach without injury during a collision. The energy that the neck bones can absorb before breaking is 8.0 J. Given the head mass of 5.0 kg, we'll find the maximum velocity using kinetic energy principles.

Calculate maximum velocity using kinetic energy

Kinetic energy is defined as \( KE = \frac{1}{2}mv^2 \). We equate this to the energy limit (8.0 J) to find the maximum velocity \( v \):\[ 8.0 = \frac{1}{2} \times 5.0 \times v^2 \]Solve for \( v \):\[ v^2 = \frac{16}{5} \]\[ v = \sqrt{3.2} \approx 1.79 \text{ m/s} \]

Convert speed from m/s to mi/h

To convert the speed from meters per second to miles per hour, use the conversion factor 1 m/s = 2.237 mi/h:\[ v_{mi/h} = 1.79 \times 2.237 \approx 4.01 \text{ mi/h} \]

Calculate acceleration of the passengers

The acceleration \( a \) can be found using the formula \( v = at \), where \( t = 0.010 \text{ s} \):\[ a = \frac{v}{t} = \frac{1.79}{0.010} = 179 \text{ m/s}^2 \]

Convert acceleration to g's

Convert the acceleration to g's using the conversion 1 g = 9.81 m/s²:\[ a_{g's} = \frac{179}{9.81} \approx 18.24 \ g \]

Calculate force acting on the head

Using Newton's second law, \( F = ma \), calculate the force:\[ F = 5.0 \times 179 = 895 \text{ N} \]

Key concepts

Whiplash

Whiplash is a common neck injury that typically occurs during a rear-end car collision. When a vehicle is abruptly hit from behind, the inertial forces cause the passengers' heads to jerk forward rapidly. This motion can lead to significant strain on the neck muscles and bones. During usual activities, muscles provide support for the bones, absorbing much of the shock. However, during a sudden collision, they may not react quickly enough, causing most of the impact burden to fall on the neck bones. The sudden movement can lead to a condition known as whiplash, where the neck experiences a rapid back-and-forth motion, similar to the cracking of a whip.

Symptoms of whiplash can vary, but often include neck pain, stiffness, and headaches. It's important to recognize these symptoms early and seek treatment, as untreated whiplash can lead to chronic pain or long-term damage. Understanding the dynamics of a collision and the resulting whiplash motion helps in designing preventative measures, such as better car seat designs and head restraints.

Collision Dynamics

Collision dynamics delves into understanding the forces during a crash. In a rear-end collision, the struck vehicle suddenly accelerates forward. This sudden acceleration can cause injuries due to the forces involved. To calculate the effects of such a collision, we use concepts like kinetic energy and acceleration.

For example, when a car is rear-ended, we calculate the maximum velocity the head can reach, ensuring it doesn't cause neck injury by using the kinetic energy formula: \[ KE = \frac{1}{2} mv^2 \]. The safety threshold for neck bones absorbing energy is limited to 8.0 J. Knowing this, the maximum speed can be determined. This speed impacts the forces felt by the body during the collision.

The acceleration experienced is another critical factor. It's calculated based on the derived speed and the time over which the collision occurs. A higher acceleration implies a stronger force, calculated through \( F = ma \). This knowledge is vital for developing safety measures in vehicles.

Neck Injury

Neck injuries during a car collision are a serious concern. The neck is particularly vulnerable due to its range of motion and vital role in connecting the head to the body. In sudden collisions, the neck could endure forces beyond its physical tolerance, leading to various injuries.

The most common neck injury in car accidents is whiplash. However, other severe injuries can include fractures or disc injuries. The human neck bones can typically resist up to 8.0 J of kinetic energy before fracturing. Therefore, during vehicular safety assessments, emphasis is placed on minimizing the energy absorbed by the neck.

Precautionary measures, such as the use of proper head restraints, can significantly reduce the risk of neck injuries. By understanding the forces at play and implementing safety technologies, the potential for severe neck injuries in collisions can be decreased, contributing to safer vehicle designs.