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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 \(8.0 \mathrm{~J}\) 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 \(70 \mathrm{~kg}\) person)? Express your answer in \(\mathrm{m} / \mathrm{s}\) and in \(\mathrm{mi} / \mathrm{h}\). (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 \(\mathrm{m} / \mathrm{s}^{2}\) and in \(\mathrm{g}\) 's.

Step by step solution

01

Calculate the maximum velocity

From the equation of kinetic energy \(KE = \frac{1}{2}mv^2\), where KE is the kinetic energy, m is the mass, and v is the velocity, we solve for v to find the maximum velocity \(v = \sqrt{\frac{2KE}{m}}\). Here, KE=8.0J (the energy that can be absorbed by the neck bones without them breaking) and m=5.0kg (the mass of the driver's head). Substitute these values into the equation to find \(v_{max}\) in m/s.

02

Convert maximum velocity to mph

To convert the maximum velocity from m/s to mph, we use the conversion factor 2.23694. Then, \(v_{max(mph)} = v_{max(m/s)} * 2.23694\)

03

Calculate the acceleration

Acceleration is the rate of change of velocity with time. So, we can calculate it as \(a = \frac{v}{t}\), where a is acceleration, v is velocity, and t is time. Here, v= \(v_{max(m/s)}\) (from part a), and t=10ms (time of collision). So plug these in to find acceleration in m/s^2.

04

Convert acceleration to g's

To convert the acceleration from m/s2 to g's, we use the conversion factor 0.101972 (since 1 g is approximately 0.101972 m/s^2). Then, \(a_{(g)} = a_{(m/s^2)} * 0.101972\)

05

Calculate the force

To calculate the force acting on the passengers' heads, we use Newton's second law of motion, which states that \(F = ma\), where F is force, m is mass, and a is acceleration. Here, m=5.0kg (the mass of the driver's head) and a=acceleration (from step 3). Plug these values into the equation to find the force in newtons.

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Key Concepts

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

Kinetic Energy Calculation

Understanding the calculation of kinetic energy (KE) is essential when examining the effects of car collisions on the human body. Kinetic energy, expressed as \( KE = \frac{1}{2}mv^2 \), quantifies the energy an object possesses due to its motion—here, \(m\) is the mass of the object and \(v\) is its velocity. In the context of car accidents, kinetic energy can be used to assess whether a force is potentially harmful.
For instance, the energy absorbed by the neck bones in a whiplash scenario must not exceed 8.0 Joules to avoid fractures. By rearranging the kinetic energy formula to solve for velocity, \( v = \sqrt{\frac{2KE}{m}} \), students can determine the greatest velocity a driver's head can reach without causing injury. In the given problem, applying this formula by substituting the maximum allowable energy for KE and the mass of an average human head demonstrates the maximum safe speed during an abrupt stop.

Physics of Car Collisions

When delving into the physics of car collisions, it's vital to decode the complex interaction of forces at play. Car collisions invoke the principles of conservation of momentum and energy, which in the simplest terms, relate to the movement and energy of the cars before and after the crash.

In a rear-end collision, as explained in the previous example, the passengers' bodies try to remain in a state of rest due to inertia while the car is suddenly pushed forward. This results in the 'whiplash' effect, where the head is snapped rapidly due to the disparity in acceleration between the head and the rest of the body. Translating this into energy terms, the concern focuses on the kinetic energy transferred to the passengers as this energy is correlated with potential injuries. The aim in safety measures and calculations is often to limit the kinetic energy and the abrupt forces acting upon the body to prevent harm.

Force and Acceleration

Newton's second law of motion forms the basis of understanding the relationship between force applied to an object and its acceleration. The law is summarized by the formula \( F = ma \), where \( F \) represents force, \( m \) is mass, and \( a \) denotes acceleration. This equation is particularly relevant when analyzing the forces that act on passengers during a car collision.
From a practical problem-solving perspective, after determining the maximum acceleration a body can tolerate without injury, like the head in a whiplash scenario, the force can then be calculated by multiplying this acceleration by the mass in question. In our exercise, the calculated acceleration, when applied over the duration of the collision, helps determine the force experienced by a passenger's head, furthering the link between the physical concepts and their consequences in real-life situations.