The spring shown in FIGURE P10.54 is compressed 50 cm and used to launch a 100 kg physics student. The track is frictionless until it starts up the incline. The student's coefficient of kinetic friction on the 30° incline is 0.15. What is the student's speed just after losing contact with the spring?
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Verified step by step guidance

1. Step 1: Identify the energy transformations involved. The spring's potential energy is converted into the student's kinetic energy as the spring releases. Use the formula for elastic potential energy: $E_{\mathrm{spring}}=\frac{1}{2}{k}_{\mathrm{spring}}{x}^2$, where $k_{\mathrm{spring}}$ is the spring constant and $x$ is the compression distance.
2. Step 2: Relate the spring's potential energy to the student's kinetic energy using the work-energy principle. The kinetic energy of the student just after losing contact with the spring is given by $E_{\mathrm{kinetic}}=\frac{1}{2}m{v}^2$, where $m$ is the mass of the student and $v$ is the speed.
3. Step 3: Equate the spring's potential energy to the student's kinetic energy to solve for the speed $v$. The equation becomes $\frac{1}{2}{k}_{\mathrm{spring}}{x}^2=\frac{1}{2}m{v}^2$. Rearrange to isolate $v$: $v=\sqrt{\frac{k_{\mathrm{spring}}}{x^2}}$.
4. Step 4: Substitute the given values into the equation. Use $m=100$ kg, $x=0.50$ m, and the spring constant $k_{\mathrm{spring}}$ (value not provided in the problem, but it must be given or calculated).
5. Step 5: Perform the algebraic simplifications to find the expression for $v$. This will yield the student's speed just after losing contact with the spring. Ensure units are consistent throughout the calculation.

Key Concepts

Hooke's Law

Hooke's Law states that the force exerted by a spring is directly proportional to its displacement from the equilibrium position, expressed as F = -kx, where k is the spring constant and x is the displacement. This principle is crucial for determining the potential energy stored in the spring when it is compressed, which is given by the formula PE = 1/2 kx².

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 is converted into kinetic energy as the student is launched, allowing us to calculate the speed just after losing contact with the spring using the equation KE = 1/2 mv².

Friction and Inclined Planes

Friction is a force that opposes motion between two surfaces in contact. On an inclined plane, the force of kinetic friction can be calculated using the coefficient of kinetic friction (μk) and the normal force. This concept is essential for understanding how the student's speed will be affected as they move up the incline, where both gravitational and frictional forces will act against their motion.