Problem 83 - University Physics with Modern Physics

Question

A student proposes a design for an automobile crash barrier in which a 1700 -kg sport utility vehicle moving at 20.0 \(\mathrm{m} / \mathrm{s}\) crashes into a spring of negligible mass that slows it to a stop. So that the passengers are not injured, the acceleration of the vehicle as it slows can be no greater than 5.00\(g .\) (a) Find the required spring constant \(k,\) and find the distance the spring will compress in slowing the vehicle to a stop. In your calculation, disregard any deformation or crumpling of the vehicle and the friction between the vehicle and the ground. (b) What disadvantages are there to this design?

Step-by-Step Solution

Answer

\(k\) is about 68,000 N/m; the spring compresses about 6.94 meters. The design might not accommodate varying speeds or weights.

Step 1: Understand the Problem

We need to find the spring constant \(k\) that would allow a 1700-kg SUV to stop without exceeding an acceleration of \(5g\), where \(g = 9.8 \ m/s^2\). We also need to find how much the spring will compress.

Step 2: Maximum Allowable Acceleration

Since the maximum allowable acceleration \(a\) is \(5g\), we calculate it as follows: \(a = 5 \times 9.8\). Thus, \(a = 49 \, \text{m/s}^2\).

Step 3: Use Work-Energy Principle

The initial kinetic energy (KE) of the vehicle is converted into the potential energy (PE) of the spring. The initial KE is given by \( \frac{1}{2} m v^2 \), where \(m\) is the vehicle's mass and \(v\) is its velocity. Set \(\frac{1}{2} m v^2 = \frac{1}{2} k x^2\), where \(x\) is the compression distance.

Step 4: Rearrange to Find \(x\) in Terms of \(k\)

Rewriting the equation from Step 3, we have \( m v^2 = k x^2 \). We need to express \(x\) using \(a\) because \(a\) is maximum acceleration.

Step 5: Newton’s Second Law

According to Newton's Second Law, the force \( F \) exerted by the spring is also \( ma = kx \). Substitute the values into this formula to find \(x\): \(1700 \times 49 = kx \).

Step 6: Relate Force and Energy

Set the potential energy (PE) of the spring \(\frac{1}{2} k x^2 = F x \). The relation \( x = \frac{mv^2}{2Fa} \) gives \(x\). Substitute \(F = ma\) from Step 5.

Step 7: Find the Spring Constant \(k\)

Now we find \(k = \frac{mv^2}{x^2}\). Use the expression for \(x\) from Step 6 and substitute for known variables to find \(k\).

Step 8: Compute \(k\) and \(x\)

Calculate the values using \(m = 1700 \, \text{kg}\), \(v = 20 \, \text{m/s}\), and \(a = 49 \, \text{m/s}^2\): Around \(k = 68,000 \, \text{N/m}\), and \(x\) is around 6.94 meters.

Step 9: Analyze Disadvantages

The spring must be long enough to compress almost 7 meters, posing size constraints and safety issues if the vehicle's speed or mass varies.

Key Concepts

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. It is directly proportional to both the mass of the object and the square of its velocity. In mathematical terms, kinetic energy (\( KE \) is calculated using the formula:

• \[ KE = \frac{1}{2} m v^2 \]

where \( m \) is the mass of the object in kilograms and \( v \) is its velocity in meters per second.
Kinetic energy is a crucial concept in understanding motion dynamics and energy conversion. In our context of the crash barrier exercise, the vehicle's kinetic energy is its initial energy due to its speed. This energy needs to be dissipated safely to bring the car to a stop. The transformation of kinetic energy into other energy forms, like potential energy in a spring, is fundamental in designing safety mechanisms.

Newton's Second Law

Newton's Second Law of Motion is one of the cornerstones of classical mechanics. It states that the force acting on an object is equal to the mass of the object multiplied by its acceleration:

• \[ F = ma \]

where \( F \) is the force in newtons, \( m \) is the mass in kilograms, and \( a \) is the acceleration in meters per second squared.
This law is vital in solving the exercise as it helps determine the force exerted by the spring on the vehicle. Since we need to ensure that the vehicle does not experience acceleration greater than a specified limit (in this case, \( 5g \)), this law provides the foundational equation to link force, mass, and acceleration. By harnessing this equation, we can identify the spring constant required to achieve the desired performance.

Work-Energy Principle

The work-energy principle is an essential concept in physics. It states that the work done on an object is equal to the change in its kinetic energy. In the context of the exercise, the principle is used to understand how the initial kinetic energy of the vehicle is converted into potential energy stored in the spring.
The equation for the principle can be written as:

• \[ \text{Work Done} = \Delta KE = KE_{final} - KE_{initial} \]

Given the vehicle comes to a complete stop, \( KE_{final} \) is zero, making the equation:

• \[ 0 = \frac{1}{2} m v^2 - \frac{1}{2} k x^2 \]

Rearranging, we find the balance between kinetic energy and the potential energy stored in the spring. This understanding is crucial to designing a crash barrier to safely dissipate energy by compressing the spring without exceeding safe acceleration limits.

Spring Constant Calculation

Calculating the spring constant \( k \) is a key part of designing a crash barrier. The spring constant reflects the stiffness of a spring: the higher the constant, the stiffer the spring.
We use the relationship derived from the equations of motion and energy:

• \[ \frac{1}{2} k x^2 = \frac{1}{2} m v^2 \]

By linking this with Newton's Second Law, where the force \( F \) is \( ma \), we find:

• \[ k = \frac{ma}{x} \]

For our exercise, with defined parameters like mass and speed, these equations guide us to calculate the spring constant. Such calculations enable us to match the spring's response to the expected forces, ensuring it compresses appropriately while safely absorbing the vehicle's kinetic energy. This evaluation highlights potential limitations, such as practicality concerns regarding the spring's size and the adaptability to changes in vehicle mass or speed.