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Chapter 2: Problem 9

Vehicle crumple zones are designed to absorb energy during an impact by deforming to reduce transfer of energy to occupants. How much kinetic energy, in Btu, must a crumple zone absorb to fully protect occupants in a 3000 -lb vehicle that suddenly decelerates from \(10 \mathrm{mph}\) to \(0 \mathrm{mph}\) ?

Short Answer

Expert verified
12.88 Btu

Step by step solution

01

- Understand the Problem

The task is to find out how much kinetic energy a crumple zone must absorb when a 3000-lb vehicle decelerates from 10 mph to 0 mph.
02

- Convert Units where Necessary

First, convert the vehicle's weight from pounds to slugs because kinetic energy calculations require mass in slugs. Use the conversion: 1 pound = 1/32.2 slug.
Vehicle mass:\[ 3000 \text{ lb} 脳 \frac{1 \text{ slug}}{32.2 \text{ lb}} = 93.17 \text{ slugs} \]
03

- Calculate Initial Kinetic Energy

Use the kinetic energy formula in its basic form: \[ KE = \frac{1}{2} mv^2 \]where m is mass in slugs and v is velocity initially given in mph. Convert 10 mph to feet per second since 1 mph = 1.467 feet/second:\[10 \text{ mph} 脳 1.467 \text{ ft/sec per mph} = 14.67 \text{ ft/sec} \] Thus, Initial KE: \[ KE = \frac{1}{2} 脳 93.17 \text{ slugs} 脳 (14.67 \text{ ft/sec})^2 = 10024.55 \text{ ft.lb} \]
04

- Convert Energy to Btu

1 Btu is equivalent to 778.17 ft.lb. So converting the kinetic energy to Btu: \[ KE = \frac{10024.55 \text{ ft.lb}}{778.17 \text{ ft.lb/Btu}} = 12.88 \text{ Btu} \]
05

Final Answer

The amount of kinetic energy the crumple zone must absorb to fully protect the occupants is 12.88 Btu.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

vehicle safety
Vehicle safety is a top priority in automotive design. Modern vehicles are equipped with crumple zones, which are areas specifically designed to absorb kinetic energy during a crash. By deforming strategically, these zones dissipate the energy that would otherwise be transferred to the vehicle鈥檚 occupants. This significantly increases the chances of survival in an accident by reducing the forces exerted on passengers. Crumple zones work alongside safety features like airbags and seatbelts to provide comprehensive protection.
unit conversion
In physics problems, converting units correctly is crucial to obtaining accurate results. This is especially important when dealing with kinetic energy, where mass and velocity need specific units. For instance, in this exercise, the vehicle's weight is given in pounds (lb), a unit that's not directly usable for kinetic energy calculations. By using the conversion factor that 1 lb is equal to 1/32.2 slugs, we convert the weight to mass suitable for the kinetic energy formula. Similarly, converting the speed from miles per hour (mph) to feet per second (ft/sec) ensures compatibility with the units used for mass and energy in the kinetic energy equation. Always keep track of your units and use appropriate conversion factors to maintain consistency throughout your calculations.
kinetic energy
demonstrates that it depends on both the mass (m) of the object and the square of its velocity (v). In our exercise, we calculated the initial kinetic energy of a vehicle by considering its mass in slugs and its velocity in feet per second. Understanding that kinetic energy increases with the square of velocity highlights why speed significantly impacts the severity of an accident. This energy must be managed effectively by vehicle design elements, such as crumple zones, to enhance passenger safety during collisions.
deceleration impact
Deceleration is the reduction of speed or velocity. During a crash, a vehicle's rapid deceleration can exert massive forces on its structure and occupants. The deceleration impact depends on the initial speed and the time taken to come to a complete stop. By calculating the kinetic energy that needs to be absorbed, we understand the extent of the impact force. Converting this energy into both pounds per foot (ft.lb) and British thermal units (Btu) provides a tangible measure to develop better safety mechanisms. These conversions help engineers design crumple zones capable of absorbing the specific amount of kinetic energy, thereby protecting passengers more effectively.

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Most popular questions from this chapter

Measured data for pressure versus volume during the compression of a refrigerant within the cylinder of refrigeration compressor are given in the table below. Using data from the table, complete the following: (a) Determine a value of \(n\) such that the data are fit by an equation of the form \(p V^{n}=\) constant. (b) Evaluate analytically the work done on the refrigerant, in Btu, using Eq. \(2.17\) along with the result of part (a). (c) Using graphical or numerical integration of the data, evaluate the work done on the refrigerant, in Btu. (d) Compare the different methods for estimating the work used in parts (b) and (c). Why are they estimates? $$ \begin{array}{ccc} \text { Data Point } & p\left(\mathrm{lbf}^{2} / \text { in. }^{2}\right) & V\left(\text { in. }^{\left.{ }^{2}\right)}\right. \\ \hline 1 & 112 & 13.0 \\ 2 & 131 & 11.0 \\ 3 & 157 & 9.0 \\ 4 & 197 & 7.0 \\ 5 & 270 & 5.0 \\ 6 & 424 & 3.0 \end{array} $$

Carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) gas within a piston-cylinder assembly undergoes a process from a state where \(p_{1}=\) \(5 \mathrm{lbf} / \mathrm{in}^{2}, V_{1}=2.5 \mathrm{ft}^{3}\) to a state where \(p_{2}=20 \mathrm{lbf} / \mathrm{in}^{2}, V_{2}=\) \(0.5 \mathrm{ft}^{3}\). The relationship between pressure and volume during the process is given by \(p=23.75-7.5 \mathrm{~V}\), where \(V\) is in \(\mathrm{ft}^{3}\) and \(p\) is in \(\mathrm{lbf} / \mathrm{in}^{2}\) Determine the work for the process, in Btu.

Air expands adiabatically in a piston-cylinder assembly from an initial state where \(p_{1}=100 \mathrm{lbf} / \mathrm{in}^{2}, v_{1}=3.704 \mathrm{ft}^{3} / \mathrm{lb}\), and \(T_{1}=1000^{\circ} \mathrm{R}\), to a final state where \(p_{2}=50 \mathrm{lbf} /\) in. \({ }^{2}\) The process is polytropic with \(n=1.4\). The change in specific internal energy, in Btu/lb, can be expressed in terms of temperature change as \(\Delta u=(0.171)\left(T_{2}-T_{1}\right)\). Determine the final temperature, in \({ }^{\circ} \mathrm{R}\). Kinetic and potential energy effects can be neglected.

The drag force, \(F_{\mathrm{d}}\), imposed by the surrounding air on a vehicle moving with velocity \(\mathrm{V}\) is given by $$ F_{\mathrm{d}}=C_{\mathrm{d}} \mathrm{A}_{2} \rho \mathrm{V}^{2} $$ where \(C_{\mathrm{d}}\) is a constant called the drag coefficient, A is the projected frontal area of the vehicle, and \(\rho\) is the air density. Determine the power, in hp, required to overcome aerodynamic drag for an automobile moving at (a) 25 miles per hour, (b) 70 miles per hour. Assume \(C_{\mathrm{d}}=0.28, \mathrm{~A}=25 \mathrm{ft}^{2}\), and \(\rho=0.075 \mathrm{lb} / \mathrm{ft}^{2}\).

For a refrigerator with automatic defrost and a topmounted freezer, the annual cost of electricity is \(\$ 55\). (a) Evaluating electricity at 8 cents per \(\mathrm{kW} \cdot \mathrm{h}\), determine the refrigerator's annual electricity requirement, in \(\mathrm{kW} \cdot \mathrm{h}\). (b) If the refrigerator's coefficient of performance is 3 , determine the amount of energy removed from its refrigerated space annually, in MJ.
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