A 500 g air-track glider moving at 0.50 m/s collides with a horizontal spring whose opposite end is anchored to the end of the track. Measurements show that the glider is in contact with the spring for 1.5 s before it rebounds. b. What is the maximum compression of the spring?

Verified step by step guidance

1. Determine the initial kinetic energy of the glider using the formula \( KE = \frac{1}{2} m v^2 \), where \( m \) is the mass of the glider and \( v \) is its velocity.
2. Assume that the kinetic energy of the glider is completely converted into potential energy of the spring at maximum compression. Use the formula for the potential energy stored in a spring, \( PE = \frac{1}{2} k x^2 \), 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 form the equation \( \frac{1}{2} m v^2 = \frac{1}{2} k x^2 \).
4. Solve for \( x \) to find the maximum compression of the spring. Rearrange the equation to \( x = \sqrt{\frac{m v^2}{k}} \).
5. To find the value of \( x \), you will need the value of the spring constant \( k \), which might be given in the problem or may need to be calculated or measured from additional data.

Key Concepts

Conservation of Momentum

The principle of conservation of momentum states that in a closed system, the total momentum before an event must equal the total momentum after the event, provided no external forces act on it. In this scenario, the momentum of the glider before it collides with the spring will be transferred to the spring and then back to the glider as it rebounds, allowing us to analyze the interaction.

Hooke's Law

Hooke's Law describes the behavior of springs, stating that the force exerted by a spring is directly proportional to its displacement from the equilibrium position, expressed as F = -kx, where F is the force, k is the spring constant, and x is the displacement. This relationship is crucial for determining the maximum compression of the spring when the glider collides with it.

Kinetic and Potential Energy

Kinetic energy is the energy of an object due to its motion, calculated as KE = 1/2 mv², where m is mass and v is velocity. When the glider compresses the spring, its kinetic energy is converted into potential energy stored in the spring, given by PE = 1/2 kx². Understanding this energy transformation is essential for calculating the maximum compression of the spring during the collision.