Problem 35 - College Physics

Question

In a simulated head-on crash test, a car impacts a wall at \(25 \mathrm{mi} / \mathrm{h}(40 \mathrm{~km} / \mathrm{h})\) and comes abruptly to rest. A 120-lb passenger dummy (with a mass of \(55 \mathrm{~kg}\) ), without a seatbelt, is stopped by an air bag, which exerts a force on the dummy of 2400 lb. How long was the dummy in contact with the air bag while coming to a stop?

Step-by-Step Solution

Answer

The dummy was in contact with the air bag for approximately 0.057 seconds.

Step 1: Understand the problem

We need to find the time it takes for the passenger dummy to come to a complete stop after hitting the air bag. Given the force (2400 lb), we will use the formula for impulse.

Step 2: Convert force to metric units

The force given is in pounds, but we need it in Newtons (N) to use it with the acceleration due to gravity. Since 1 lb is approximately 4.44822 N, the force in Newtons is \(2400 \times 4.44822 = 10675.728 \text{ N}\).

Step 3: Convert velocity to metric units

The car's initial velocity is given in miles per hour. We convert this to meters per second: \(25 \text{ mi/h} = 40 \text{ km/h} = \frac{40 \times 1000}{3600} \text{ m/s} \). The dummy's initial speed is \(11.11 \text{ m/s}\).

Step 4: Apply the impulse-momentum principle

According to the impulse-momentum principle, impulse (force times time) equals change in momentum \( (m \Delta v) \). Thus, \( F \Delta t = m \Delta v \), where \(m = 55 \text{ kg}\) is the dummy's mass, and \(\Delta v = 11.11 \text{ m/s}\).

Step 5: Solve for time \(\Delta t\)

Rearrange the formula to solve for the time: \(\Delta t = \frac{m \Delta v}{F} = \frac{55 \times 11.11}{10675.728} \approx 0.057 \text{ seconds}\).

Key Concepts

Impulse

The concept of impulse is fundamental in understanding how forces affect the motion of objects. Impulse can be understood as the product of force applied to an object and the time duration over which this force acts. In mathematical terms, impulse is calculated by the equation \( J = F \times \Delta t \), where \( J \) is impulse, \( F \) is the force, and \( \Delta t \) is the time.
An impulse causes a change in momentum of an object. This is why impulse is critical in collision scenarios, such as a car crash, where rapid decelerations occur. Situations like the air bag interaction in a crash test demonstrate how impulse works to safely absorb energy and minimize injury.

Momentum

Momentum is a measure of the motion of an object, defined as the product of its mass and velocity. The formula for momentum \( p \) is \( p = m \times v \), where \( m \) is the mass of the object and \( v \) is its velocity. In scenarios of collision, such as with the dummy in the car crash test, momentum plays a critical role in understanding how and why forces act.
When the dummy hits the airbag, its momentum changes rapidly, hence the need to calculate the impulse. Since an object will have momentum in the direction of its velocity, in this exercise, the dummy's initial momentum needs to be offset by the airbag's force to bring it to rest. By optimizing the change in momentum, the force exerted by the airbag can prevent greater injury.

Force Conversion

Force conversion is often necessary in physics problems to ensure that units are consistent, enabling equations to be used correctly. In this exercise, the force exerted by the airbag needs to be converted from pounds to Newtons. This is because the standard unit of force in the International System of Units (SI) is the Newton.
The conversion factor for pounds to Newtons is approximately \(1 \text{ lb} = 4.44822 \text{ N}\). Therefore, a force of \(2400 \text{ lb}\) equates to \(2400 \times 4.44822 \approx 10675.73 \text{ N}\). This conversion allows us to integrate with other SI units such as meters and kilograms, ensuring accurate calculations of impulse and motion.

Velocity Conversion

Velocity conversion is crucial when working with different unit systems. In the described exercise, the car's velocity is initially given in miles per hour, which needs to be converted into meters per second for consistency with other SI units.
The conversion process involves recognizing that \(1 \text{ mile} = 1.60934 \text{ km}\) and dividing the result over the number of seconds in an hour. For the given velocity, \(25 \text{ mi/h}\) converts to \(40 \text{ km/h}\), and then \(\frac{40 \times 1000}{3600}\) gives a velocity of \(11.11 \text{ m/s}\). Knowing this conversion is vital as it ties directly into the calculations of the dummy’s momentum and ultimately the impulse-momentum principle.