Answer :
The coefficient of friction between the ice and the skates is 0.036.
To explain this answer, we will first analyze the problem and then derive the formula to calculate the coefficient of friction.
In this problem, we are given the initial velocity of the ice skater (13.7 m/s) and the distance the skater travels before coming to a halt (97.6 m). We are asked to find the coefficient of friction between the ice and the skates. The hint suggests that there might be a crash or sudden stop involved, which indicates that the force of friction plays a significant role in this situation.
To approach this problem, we can use the formula for the work done by the frictional force (F_f) acting on an object:
Work = F_f × Distance
In this case, the work done by the frictional force is equal to the change in the ice skater's kinetic energy, which is converted into thermal energy due to friction. The formula for kinetic energy is:
Kinetic Energy = 0.5 × Mass × (Initial Velocity)^2
Since we are considering the coefficient of friction (μ), we can rewrite the formula for the work done by the frictional force as:
Work = μ × Normal Force × Distance
As the ice surface is smooth, the coefficient of friction remains constant during the process. We can now equate the work done by the frictional force to the change in the ice skater's kinetic energy:
0.5 × Mass × (Initial Velocity)^2 = μ × Normal Force × Distance
We can now plug in the given values:
0.5 × Mass × (13.7 m/s)^2 = μ × Normal Force × 97.6 m
Since the mass of the ice skater is not given, we can assume it to be 'm'. Now, we can rewrite the formula for kinetic energy as:
0.5 × m × (13.7 m/s)^2 = μ × m × g × 97.6 m
Here, 'g' represents the acceleration due to gravity (approximately 9.81 m/s²). Now, we can cancel out the mass (m) from both sides:
0.5 × (13.7 m/s)^2 = μ × g × 97.6 m
Solving the equation for μ, we get:
μ = (0.5 × (13.7 m/s)^2) / (g × 97.6 m)
μ ≈ 0.036
Hence, the coefficient of friction between the ice and the skates is approximately 0.036. This value indicates that the friction between the ice and the skates is relatively low, allowing the ice skater to coast at a high speed before coming to a halt. This low coefficient of friction is essential for smooth and efficient ice skating performances.
In conclusion, we have derived the coefficient of friction between the ice and the skates by analyzing the given problem and using the appropriate formulas. The final answer is 0.036, which signifies the effective conversion of the ice skater's kinetic energy into thermal energy due to friction. This low coefficient of friction allows the ice skater to move at high speeds and maintain control while performing various maneuvers on the smooth ice surface.