Problem 10 - College Physics

Question

A man claims he can safely hold on to a \(12.0-\mathrm{kg}\) child in a head-on collision with a relative speed of \(120-\mathrm{mi} / \mathrm{h}\) lasting for \(0.10 \mathrm{~s}\) as long as he has his seat belt on. (a) Find the magnitude of the average force needed to hold onto the child. (b) Based on the result to part (a), is the man's claim valid? (c) What does the answer to this problem say about laws requiring the use of proper safety devices such as seat belts and special toddler seats?

Step-by-Step Solution

Answer

The average force required to hold onto the child during the head-on collision is 6436.8 N. It is unlikely that a man, even wearing a seat belt, would be able to exert this force, challenging the validity of his claim. This problem emphasizes the importance of laws mandating the use of safety devices like seat belts and special toddler seats in vehicles.

Step 1: Convert the speed from miles per hour to meters per second

The relative speed given, 120 mi/h, must be converted to m/s for the purpose of using coherent SI units. To do this, use the conversion factor \(1 \mathrm{mi/h} = 0.44704 \mathrm{m/s}\). So, the speed in m/s is \(120 \mathrm{mi/h} \times 0.44704 \mathrm{m/s}\) = \(53.64 \mathrm{m/s}\).

Step 2: Calculate the acceleration

Acceleration is the change in velocity divided by the time it takes for the change to occur. In this case the final velocity after the collision is zero (because the child is held still), and the initial velocity is the speed the car was moving with, thus the change in velocity is \(-53.64 \mathrm{m/s}\). The time during which the change occurs is 0.10s. So, the acceleration is \(-53.64 \mathrm{m/s} / 0.10 \mathrm{s} = -536.4 \mathrm{m/s}^{2}\). The negative sign indicates a decrease in velocity, or deceleration.

Step 3: Calculate the Force

Now that the acceleration value is known, Newton's second law of motion can be used to calculate the force. The law states that force equals mass times acceleration (F = ma). Substituting the mass of the child (12.0 kg) and the acceleration (-536.4 m/s²), the force exerted is \(F = 12.0 \mathrm{kg} \times -536.4 \mathrm{m/s}^{2} = -6436.8 \mathrm{N}\). The minus sign shows that the force is exerted opposite to the direction of motion. The magnitude of this force is 6436.8 N.

Step 4: Evaluate the claim and discuss safety measures

Given the computed force required to hold the child still during the collision, it's improbable that the man's claim is valid. Even with his seat belt on, exerting about 6436.8 N of force to maintain a grip on the child seems unlikely. Considering such scenarios, laws requiring the use of proper safety devices such as seat belts and special toddler seats are crucial. These tools are engineered to withstand high impacts and provide safety beyond human limitations.

Key Concepts

Newton's second law of motion

At the heart of classical mechanics lies Newton's second law of motion, which describes the relationship between the motion of an object and the forces acting upon it. This law can be succinctly stated as F = ma, where F represents force, m is mass, and a is acceleration. In practice, this means that the force exerted on an object is equal to its mass multiplied by the acceleration it experiences. For instance, when a parent tries to hold a child stationary during a collision, the force required is determined by the child's mass and how quickly the child's speed changes, or decelerates, during the crash.

Understanding this fundamental principle is critical, especially when evaluating scenarios such as a car collision. The law implies that an increase in either mass or acceleration results in a proportional increase in force, which underscores the necessity for safety restraint systems that can manage the substantial forces during automobile accidents. So, while a man may claim he can hold onto a child in a collision, the immense forces calculated using Newton's second law suggest that this would be extraordinarily difficult, if not impossible, without additional support, like a proper child safety seat.

Conversion of units

When working with physics problems, particularly those involving equations of motion, it's essential to maintain consistency in units. In the initial problem, speed is provided in miles per hour (mi/h) but for calculation purposes, it must be converted to meters per second (m/s), which is the standard unit of speed in the International System of Units (SI).

The process of converting units involves using conversion factors, which are fixed values that describe how one unit of measure relates to another. In this case, knowing that 1 mi/h is equivalent to 0.44704 m/s allows us to convert 120 mi/h into an SI-compliant speed. This conversion is a crucial step that ensures accuracy in subsequent calculations and reveals the sometimes hidden magnitude of physical phenomena, like the severity of forces involved in automotive collisions.

Deceleration

Deceleration is the term used to describe a decrease in speed or negative acceleration. In our exercise, the child is brought to an abrupt halt from a speed of 53.64 m/s over a brief 0.10-second interval during a head-on collision. Deceleration is determined by the rate of change in velocity, measured as the difference in speed over time.

Mathematically, this deceleration rate is substantial, amounting to -536.4 m/s². In real-world terms, such rapid deceleration amounts to experiencing extreme forces. In traffic accidents, these forces can lead to serious injury, which is why slowing down impact events with crumple zones, airbags, and safety belts is vital. Deceleration helps frame the critical need for these safety systems, particularly when protecting vulnerable passengers such as children.

Child safety during collisions

The safety of child passengers during vehicle collisions is a topic of paramount concern. Children have unique physiological qualities that make them more susceptible to injury in the event of sudden stops or crashes. Therefore, it's essential to use appropriate child safety seats, which are designed to secure and protect children based on their size and weight.

As illustrated by the force calculation exercise, the average force required to hold a child during a collision can be tremendously high - far exceeding the capability of a person holding the child, even one wearing a seatbelt. Child safety seats are engineered to distribute these forces over a larger area of the child's body, significantly reducing the risk of injury. Laws mandate their use because they are based on extensive research and real-world evidence, demonstrating that proper restraints significantly improve child safety outcomes during vehicle collisions.