1. Convert Units
* Speed: 95 kph = 26.39 m/s (1 kph = 1000 m / 3600 s)
* Acceleration: 4.0 g = 39.2 m/s² (g = 9.8 m/s²)
2. Understand the Concepts
* Work-Energy Principle: The work done by the spring on the car will equal the change in kinetic energy of the car.
* Spring Potential Energy: The potential energy stored in a spring is given by (1/2)kx², where k is the spring constant and x is the compression distance.
* Kinetic Energy: The kinetic energy of an object is given by (1/2)mv², where m is the mass and v is the velocity.
3. Set up the Equations
* Work-Energy: (1/2)kx² = (1/2)mv²
* Acceleration: a = k/m * x (Since a = F/m and F = kx)
4. Solve for the Spring Constant (k)
* From the acceleration equation: x = (a*m)/k
* Substitute x in the work-energy equation: (1/2)k[(a*m)/k]² = (1/2)mv²
* Simplify and solve for k: k = (m * v²) / (a * x)
5. Calculate the Compression Distance (x)
* We need to find the compression distance 'x' to proceed. We can use the acceleration equation:
* x = (a * m) / k
* Since we don't know 'k' yet, we'll need to use another approach to find 'x'.
* Consider the stopping distance: Assume the car comes to a complete stop after compressing the spring. We can use the following kinematic equation:
* v² = u² + 2as
* Where:
* v = final velocity (0 m/s)
* u = initial velocity (26.39 m/s)
* a = acceleration (-39.2 m/s²)
* s = stopping distance (x)
* Solve for x: x = (v² - u²) / (2a) = (0² - 26.39²) / (2 * -39.2) ≈ 8.87 m
6. Calculate the Spring Constant (k)
* Now that we have the compression distance 'x', we can calculate the spring constant:
* k = (m * v²) / (a * x)
* k = (1200 kg * (26.39 m/s)²) / (39.2 m/s² * 8.87 m)
* k ≈ 2152 N/m
Therefore, the spring constant k should be approximately 2152 N/m to bring the 1200 kg car to rest from 95 kph with a maximum acceleration of 4.0 g.
Important Note: This calculation assumes the spring is designed to act as the sole means of stopping the car. In a real-world scenario, other factors like crumple zones and safety features would also contribute to the stopping process. This solution provides a theoretical estimate.
