Problem 17 - College Physics

Question

\(\bullet$$\bullet\) Air-bag safety. According to safety standards for air bags, the maximum acceleration during a car crash should not exceed 60\(g\) and should last for no more than 36 ms. (a) In such a case, what force does the air bag exert on a 75 kg person? Start with a free-body diagram. (b) Express the force in part (a) in terms of the person's weight.

Step-by-Step Solution

Answer

The airbag exerts a force of 44145 N, which is about 60 times the person's weight.

Step 1: Understanding the Problem

The problem involves computing the force exerted by an airbag on a 75 kg person during a car crash, where the maximum allowed acceleration is 60 times gravitational acceleration \(g\). You'll also need to express this force relative to the person's weight.

Step 2: Free-Body Diagram Analysis

Imagine the free-body diagram for the person: The airbag exerts an upward force on the person during the crash to decelerate them. Assuming negligible other forces for simplicity, this force is the key to finding the answer.

Step 3: Calculate the Force Using Maximum Acceleration

The acceleration involved is \(a = 60g\), where \(g = 9.81 \, \text{m/s}^2\). Use Newton's second law, \(F = ma\), where \(m = 75 \, \text{kg}\). Thus, the force \(F\) is given by:\[F = 75 \, \text{kg} \times 60 \times 9.81 \, \text{m/s}^2\]

Step 4: Solve for Force

Substituting the values, compute the force:\[F = 75 \, \text{kg} \times 588.6 \, \text{m/s}^2 = 44145 \, \text{N}\]Thus, the airbag exerts a force of 44145 N on the person.

Step 5: Express Force as a Multiple of Weight

The person's weight is given by \(W = mg = 75 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 735.75 \, \text{N}\). Now express the force as a multiple of this weight:\[\text{Force as multiple of weight} = \frac{44145}{735.75} \]

Step 6: Calculate the Multiple

Divide the force by the weight to find the ratio:\[\frac{44145}{735.75} \approx 60\]The force is approximately 60 times the person's weight.

Key Concepts

Airbag Safety Standards

Airbags are essential for car safety, designed to minimize injury during collisions. Safety standards set shared norms for their performance. The maximum acceleration that airbags should allow is 60 times the force of gravity. This is to ensure a rapid but controlled deceleration of the passengers. Such intense deceleration must last no longer than 36 milliseconds.
Having a standard like 60\(g\) (where \(g = 9.81 \, \text{m/s}^2\)) helps manufacturers design systems to cushion the passengers effectively. Airbags need to deploy swiftly but also safely to absorb crash forces without causing additional harm.
These standards aim to balance effective deceleration with minimizing risk to the human body, which can be vulnerable to rapid changes in speed.

Free-Body Diagram

A free-body diagram simplifies understanding of forces in physics problems. It is a visual tool to identify and represent all forces acting on an object.
In the context of an airbag during a crash, a free-body diagram for the person would typically include:

• A force exerted upwards by the airbag, opposing the direction of motion.
• Neglecting other forces allows focusing on this dominant force during the crash.

By isolating these forces, one can easily apply Newton's Second Law of Motion (\(F = ma\)). This helps calculate the force an airbag exerts to bring a passenger to a stop.

Force Calculation During Car Crash

Calculating the force experienced during a car crash involves understanding mass, acceleration, and Newton's Second Law. The law states that force equals mass times acceleration (\(F = ma\)).
Consider a 75 kg person experiencing acceleration of 60\(g\). Here, \(g\) is approximately \(9.81 \, \text{m/s}^2\), making the total acceleration \(60 \times 9.81\).
Plug these values into the equation:\[F = 75 \, \text{kg} \times 60 \times 9.81 \, \text{m/s}^2 = 44145 \, \text{N}.\]This calculates the force exerted by the airbag in newtons. For comparison, you can express this force as multiples of the person's weight, where weight \(W\) is \(m \times g = 735.75 \, \text{N}\). Thus, \[\text{Force as a multiple of weight} = \frac{44145}{735.75} \approx 60.\]This shows the airbag exerts a force 60 times the person's weight, illustrating how airbags manage extreme forces in crashes.