An energy-absorbing car bumper has a spring constant of 410 kN/m. Find the maximum compression of the bumper if the car, with mass 1300 kg, collides with a wall at a speed of 2.0 m/s (approximately 5 mi/h ).

Verified step by step guidance

1. Identify the initial kinetic energy of the car using the formula for kinetic energy, KE = \frac{1}{2}mv^2, where m is the mass of the car and v is its velocity.
2. Recognize that the kinetic energy of the car will be converted into potential energy stored in the compression of the spring, given by the formula for elastic potential energy, PE = \frac{1}{2}kx^2, where k is the spring constant and x is the compression of the spring.
3. Set the kinetic energy equal to the potential energy to create an equation: \frac{1}{2}mv^2 = \frac{1}{2}kx^2.
4. Solve for x, the compression of the spring, by rearranging the equation to isolate x: x = \sqrt{\frac{mv^2}{k}}.
5. Plug in the values for m (1300 kg), v (2.0 m/s), and k (410,000 N/m) into the equation to find the maximum compression x.

Key Concepts

Spring Constant

The spring constant, denoted as 'k', measures the stiffness of a spring. It is defined as the force required to compress or extend the spring by a unit distance. In this case, a spring constant of 410 kN/m indicates that a force of 410,000 Newtons is needed to compress the spring by one meter. This concept is crucial for understanding how much the bumper will compress upon impact.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion, calculated using the formula KE = 0.5 * m * v², where 'm' is mass and 'v' is velocity. In this scenario, the car's kinetic energy before the collision will be converted into potential energy stored in the compressed bumper. Understanding this energy transformation is essential for determining how much the bumper compresses.

Energy Conservation

The principle of energy conservation states that energy cannot be created or destroyed, only transformed from one form to another. In the context of the car bumper collision, the kinetic energy of the moving car is converted into elastic potential energy in the spring when the bumper compresses. This concept is fundamental for solving the problem, as it allows us to equate the initial kinetic energy to the potential energy at maximum compression.