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Ch 06: Dynamics I: Motion Along a Line
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 06: Dynamics I: Motion Along a LineProblem 42a
Chapter 6, Problem 42a

Seat belts and air bags save lives by reducing the forces exerted on the driver and passengers in an automobile collision. Cars are designed with a 'crumple zone' in the front of the car. In the event of an impact, the passenger compartment decelerates over a distance of about 1 m as the front of the car crumples. An occupant restrained by seat belts and air bags decelerates with the car. By contrast, an unrestrained occupant keeps moving forward with no loss of speed (Newton's first law!) until hitting the dashboard or windshield. These are unyielding surfaces, and the unfortunate occupant then decelerates over a distance of only about 5 mm. A 60 kg person is in a head-on collision. The car's speed at impact is 15 m/s. Estimate the net force on the person if he or she is wearing a seat belt and if the air bag deploys.

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Step 1: Identify the given values in the problem. The mass of the person is \( m = 60 \, \text{kg} \), the initial velocity of the car is \( v_i = 15 \, \text{m/s} \), and the final velocity after the collision is \( v_f = 0 \). The deceleration distance for a restrained occupant is \( d = 1 \, \text{m} \).
Step 2: Use the kinematic equation to calculate the acceleration (deceleration) experienced by the person: \( v_f^2 = v_i^2 + 2ad \). Rearrange the equation to solve for \( a \): \( a = \frac{v_f^2 - v_i^2}{2d} \). Substitute the known values into the equation.
Step 3: Once the acceleration \( a \) is determined, use Newton's second law of motion to calculate the net force acting on the person: \( F = ma \). Substitute the mass of the person and the calculated acceleration into this equation.
Step 4: Ensure that the units are consistent throughout the calculations. The velocity is in meters per second (m/s), the distance is in meters (m), and the mass is in kilograms (kg). The resulting force will be in newtons (N).
Step 5: Interpret the result. The calculated force represents the net force exerted on the person due to the deceleration provided by the seat belt and air bag. This force is significantly less than what would occur if the person were unrestrained, highlighting the importance of these safety features.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Newton's First Law of Motion

Newton's First Law states that an object at rest will remain at rest, and an object in motion will continue in motion at a constant velocity unless acted upon by a net external force. In the context of a car collision, an unrestrained occupant continues moving forward due to inertia, leading to severe injuries upon impact with the dashboard or windshield. This principle highlights the importance of seat belts and airbags in changing the motion of the occupant safely.
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Crumple Zones

Crumple zones are designed areas in a vehicle that deform and absorb energy during a collision, thereby reducing the force transmitted to the occupants. By allowing the car to decelerate over a longer distance (about 1 meter), crumple zones help to minimize the impact forces experienced by passengers. This design feature is crucial for enhancing safety in the event of an accident.
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Force and Deceleration

The net force experienced by an occupant during a collision can be calculated using Newton's second law, F = ma, where 'F' is the force, 'm' is the mass, and 'a' is the acceleration (or deceleration). When a restrained occupant decelerates over a longer distance due to seat belts and airbags, the average force exerted on them is significantly lower compared to an unrestrained occupant, who decelerates abruptly over a much shorter distance, resulting in a higher force and greater risk of injury.
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Related Practice
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