You work for a toy company, and you're designing a springlaunched model rocket. The launching apparatus has room for a spring that can be compressed \(14 \mathrm{cm},\) and the rocket's mass is \(65 \mathrm{g} .\) If the rocket is to reach an altitude of \(35 \mathrm{m},\) what should you specify for the spring constant?

The principle of conservation of energy is a foundational concept in physics, asserting that energy cannot be created or destroyed, only transformed from one form to another. In the context of our spring-launched model rocket, conservation of energy plays a crucial role.

When the spring is compressed, it stores energy in the form of elastic potential energy. As the spring is released and the rocket is launched, this potential energy is converted into kinetic energy, which propels the rocket upwards. When the rocket reaches its highest point, the kinetic energy has been transformed into gravitational potential energy. According to conservation of energy, the energy initially stored in the spring should be equal to the gravitational potential energy when the rocket reaches the desired altitude, allowing us to equate them and find the necessary spring constant for the desired launch height.

Understanding this concept is essential because it ensures that the system's energy before and after the rocket is launched remains constant, providing the basis to solve for the unknown spring constant.

Potential energy is the energy that is stored within an object due to its position, state, or configuration. In the case of our model rocket, there are two types of potential energy to consider: elastic potential energy in the compressed spring and gravitational potential energy at the peak of the rocket's flight.

The formula for elastic potential energy, derived from Hooke's Law, is \( \frac{1}{2}kx^2 \) where \( k \) represents the spring constant and \( x \) the compression distance of the spring. Gravitational potential energy, on the other hand, is described by the equation \( mgh \) where \( m \) is the mass of the object, \( g \) is the acceleration due to gravity, and \( h \) is the height above a reference point - in this case, the height the rocket is to reach.

By comprehending these different forms of potential energy, students can understand how the energy stored in a spring can be utilized to launch an object to a certain height based on the properties of the spring and the mass of the object.

Hooke's Law is a principle that describes the behavior of springs subjected to force. It states that the force needed to extend or compress a spring by some distance scales linearly with that distance. The law is mathematically expressed as \( F = -kx \) where \( F \) is the force applied, \( k \) is the spring constant, and \( x \) is the distance the spring is stretched or compressed from its equilibrium position.

The negative sign in the formula indicates that the force exerted by the spring is in the opposite direction of the force being applied - this is known as restoring force. For our toy rocket, Hooke's Law helps us calculate the elastic potential energy stored in the spring when compressed, which is essential in determining how powerful the spring must be to launch the rocket to the required height. The relationship given by Hooke's Law is utilized in the energy equation for the elastic potential energy of the spring set to the gravitational potential energy at the rocket's peak altitude, thus facilitating the calculation of the spring constant.