Problem 88 - College Physics

Question

\(\bullet\) \(\bullet\) \(\bullet\) Automobile air-bag safety. An automobile air bag cush- ions the force on the driver in a head-on collision by absorbing her energy before she hits the steering wheel. Such a bag can be modeled as an elastic force, similar to that produced by a spring. (a) Use energy conser- vation to show that the effec- tive force constant \(k\) of the air bag is \(k=m v_{0}^{2} / x_{\text { max }}^{2},\) where \(m\) is the mass of the driver, \(v_{0}\) is the speed of the car (and driver) at the instant of the accident, and \(x_{\max }\) is the maxi- mum distance the bag gets compressed, which, in a severe accident, would be the dis- tance from the driver's body to the steering wheel. (b) Show that the maximum force the air bag would exert on the driver is \(F_{\max }=m v_{0}^{2} / x_{\max }\) (c) Now let's put in some realistic numbers. Experimental tests have shown that injury occurs when a force density greater than \(5.0 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}\) acts on human tissue. (The total force is this force density times the area over which it acts. As the accom- panying figure shows, the force of the air bag acts mostly on the upper front half of the driver's body, over an area of about 2500 \(\mathrm{cm}^{2} .\) (Check your own body to see if this is reasonable.) Use this value to calculate the total force on the driver's body at the threshold of injury. (d) Use your results to calculate the effective force constant \(k\) of the air bag and the maximum speed for which the bag will prevent injury to a 65 kg driver if she is 30 \(\mathrm{cm}\) from the steering wheel at the instant of impact. Express the speed in \(\mathrm{m} / \mathrm{s}\) and mph. (e) How could you design a safer air bag for higher speed collisions? What things could you alter to do this? Would it be safe to make a stiffer air bag by inflating it more? Explain your reasoning.

Step-by-Step Solution

Answer

Derived formulas show safer designs involve increasing air bag compression. A 65 kg driver is safe under 18.5 m/s, or 41.4 mph.

Step 1: Understanding the Problem and Energy Conservation in Air Bags

We need to use energy conservation to relate the kinetic energy of the driver at the moment of the crash to the potential energy stored in the airbag when it's fully compressed. Initially, the kinetic energy is given by \( \frac{1}{2} m v_0^2 \), and by the time the driver compresses the airbag completely, all this kinetic energy is transformed into elastic potential energy, \( \frac{1}{2} k x_{\text{max}}^2 \). Setting these equal allows us to solve for the force constant \( k \).

Step 2: Derive the Effective Force Constant

By using the energy conservation principle, equate the kinetic energy with the potential energy: \( \frac{1}{2} m v_0^2 = \frac{1}{2} k x_{\text{max}}^2 \). Solving for \( k \), we find \( k = \frac{m v_0^2}{x_{\text{max}}^2} \). This formula relates the mass of the driver, velocity at impact, and the maximum displacement of the airbag.

Step 3: Determine the Maximum Force Exerted by the Air Bag

The maximum force exerted by the airbag can be found from the force constant \( k \) and the maximum compression distance \( x_{\text{max}} \), using the formula for the spring force \( F_{\text{max}} = k x_{\text{max}} \). Substitute \( k = \frac{m v_0^2}{x_{\text{max}}^2} \) into this, resulting in \( F_{\text{max}} = m v_0^2 / x_{\text{max}} \).

Step 4: Calculate Total Force on Driver at Threshold of Injury

Given a force density threshold of \( 5.0 \times 10^5 \text{ N/m}^2 \) and an effective area of impact \( 2500 \text{ cm}^2 = 0.25 \text{ m}^2 \), calculate the total force as \( F = ext{Force density} \times ext{Area} = 5.0 \times 10^5 ext{ N/m}^2 \times 0.25 ext{ m}^2 = 1.25 \times 10^5 ext{ N} \).

Step 5: Calculate the Force Constant and Maximum Safe Speed

To avoid injury to a 65 kg driver at an initial distance of 0.3 m from the wheel, set \( F_{\text{max}} = 1.25 \times 10^5 \text{ N} \). Using \( F_{\text{max}} = m v_0^2 / x_{\text{max}} \), solve for \( v_0 \) as \( v_0 = \sqrt{\frac{F_{\text{max}} \cdot x_{\text{max}}}{m}} \). Plugging in values gives \( v_0 = \sqrt{\frac{1.25 \times 10^5 \cdot 0.3}{65}} \approx 18.5 \text{ m/s} \). To convert to mph, use 1 m/s = 2.237 mph, which results in approximately 41.4 mph.

Step 6: Suggestions for a Safer Air Bag Design

To design a safer air bag for high-speed collisions, consider increasing the maximum compression distance \( x_{\text{max}} \), which would decrease \( F_{\text{max}} \) for a given speed, or use materials that distribute forces more evenly over the driver's body. Making the airbag stiffer by inflating it more might actually increase \( F_{\text{max}} \) at any given displacement, potentially increasing the risk of injury.

Key Concepts

Energy Conservation

Energy conservation is a fundamental concept in physics, particularly relevant in the context of automobile air bags. When a car crashes, the kinetic energy of the moving vehicle and driver must go somewhere. It transforms primarily into potential energy as the air bag inflates and absorbs the energy. This change is crucial because it reduces the force experienced by the driver during a collision.

Here's how it works: At the moment of impact, the driver's kinetic energy is \( \frac{1}{2} m v_0^2 \), where \( m \) is the mass of the driver and \( v_0 \) is the speed just before the impact. Each joule of this kinetic energy is transferred into the air bag, helping it to compress and store what we call elastic potential energy, defined by the formula \( \frac{1}{2} k x_{\text{max}}^2 \).

Setting these two energies equal: \( \frac{1}{2} m v_0^2 = \frac{1}{2} k x_{\text{max}}^2 \), we can derive useful information about the forces involved in cushioning the driver during a collision.

Force Constant

In the context of air bag safety, the force constant \( k \) plays a critical role. This constant characterizes how much force an airbag can exert per unit of compression. It is akin to the stiffness of a spring.

By using the derived equation \( k = \frac{m v_0^2}{x_{\text{max}}^2} \), we see how the force constant \( k \) depends on the driver's mass \( m \), the speed \( v_0 \), and the maximum compression of the airbag, \( x_{\text{max}} \).

This relationship shows that, for safer car designs, engineers might adjust the airbag's material or structural properties to achieve an optimal force constant, ensuring enough energy absorption while minimizing force transmitted to the driver.

Air Bag Design

The design of an automobile air bag is crucial for ensuring passenger safety in a collision. Air bags must be carefully engineered to absorb kinetic energy effectively and distribute forces over a wide area of the driver's body.

Effective air bag designs consider:

• Compression Distance: Increasing \( x_{\text{max}} \) or the maximum stretching of the air bag can result in less force on the driver by giving more room and time to decelerate safely.
• Force Distribution: Designing airbags to apply pressure over a larger surface helps lower the concentrated force on any one part of the driver’s body.
• Material Choice: Using materials that offer a controlled deployment and compression can help manage the energy transfer more effectively without causing additional harm.

This design approach can significantly affect how force fields travel through human tissue and determine the success in achieving a gentle, even deceleration during a crash.

Kinetic Energy

Kinetic energy is the energy of motion that a moving object, including drivers, possesses. In the realm of vehicle safety, understanding kinetic energy helps in mitigating crash effects.

In a crash scenario, the kinetic energy of a driver \( \frac{1}{2} m v_0^2 \) is what the airbag seeks to neutralize. The velocity \( v_0 \) is particularly important as this is squared in the kinetic energy equation, indicating how increased speeds dramatically heighten the energy.

By converting kinetic energy into other forms, such as potential energy of a compressed airbag, modern safety systems are designed to reduce the risk of injury, protecting the passenger by ensuring a controlled slowdown.

Potential Energy

Potential energy in an airbag is akin to a spring being compressed. It is part of the energy transformation process that makes air bags effective safety devices. As kinetic energy from a moving driver is absorbed, the airbag compresses, storing this energy as elastic potential energy.

The potential energy is vital for understanding how air bags cushion impacts. It is mathematically represented as \( \frac{1}{2} k x_{\text{max}}^2 \), where \( k \) is the force constant and \( x_{\text{max}} \) is how much the airbag compresses.

By allowing this energy to gradually return to the driver, air bags act as a buffer, slowly decelerating the force experienced during a crash. Thus, a well-designed airbag efficiently transforms kinetic energy into potential energy, and then back again, all to protect the human body from the direct impacts of a collision.