You have been hired to design a spring-launched roller coaster that will carry two passengers per car. The car goes up a 10-m-high hill, then descends 15 m to the track's lowest point. You've determined that the spring can be compressed a maximum of 2.0 m and that a loaded car will have a maximum mass of 400 kg. For safety reasons, the spring constant should be 10% larger than the minimum needed for the car to just make it over the top. What is the maximum speed of a 350 kg car if the spring is compressed the full amount?

Verified step by step guidance

1. Step 1: Analyze the problem and identify the key concepts. This problem involves energy conservation, specifically the conversion of elastic potential energy stored in the spring into gravitational potential energy and kinetic energy. The spring constant is given as 10% larger than the minimum required for the car to reach the top of the hill.
2. Step 2: Calculate the minimum spring constant required for the car to reach the top of the hill. Use the energy conservation principle: the elastic potential energy of the spring is converted into gravitational potential energy at the top of the hill. The formula for elastic potential energy is \( U_s = \frac{1}{2} k x^2 \), and the formula for gravitational potential energy is \( U_g = m g h \). Set \( U_s \geq U_g \) and solve for \( k \): \( k = \frac{2 m g h}{x^2} \). Then, increase this value by 10% for safety.
3. Step 3: Determine the total energy stored in the spring when compressed the full amount. Use the formula for elastic potential energy: \( U_s = \frac{1}{2} k x^2 \), where \( k \) is the spring constant (including the 10% safety margin) and \( x \) is the maximum compression of the spring (2.0 m).
4. Step 4: Apply the energy conservation principle to find the maximum speed of the car. At the lowest point of the track, all the elastic potential energy stored in the spring is converted into kinetic energy. The formula for kinetic energy is \( K = \frac{1}{2} m v^2 \). Set \( U_s = K \) and solve for \( v \): \( v = \sqrt{\frac{2 U_s}{m}} \). Substitute \( U_s \) and \( m \) (350 kg) into the equation.
5. Step 5: Perform the calculations using the values determined in the previous steps to find the maximum speed of the car. Ensure all units are consistent (e.g., meters, kilograms, seconds) and verify the result.

Key Concepts

Potential Energy

Potential energy is the energy stored in an object due to its position in a gravitational field. In the context of the roller coaster, as the car ascends the 10-meter hill, it gains gravitational potential energy, which can be calculated using the formula PE = mgh, where m is mass, g is the acceleration due to gravity, and h is the height. This energy is crucial for understanding how the car can reach the top of the hill.

Spring Potential Energy

Spring potential energy is the energy stored in a compressed or stretched spring, described by Hooke's Law. The formula for spring potential energy is PE_spring = 1/2 kx², where k is the spring constant and x is the compression or extension of the spring. This concept is essential for determining how much energy is available to propel the roller coaster car when the spring is fully compressed.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In this scenario, the potential energy stored in the compressed spring will convert into kinetic energy as the car is launched. Understanding this principle allows us to calculate the maximum speed of the car at the lowest point of the track by equating the spring potential energy to the kinetic energy of the car.