91Ó°ÊÓ

The coefficient of friction (bc) is a number that represents the degree of interaction between two surfaces. It varies from one material combination to another. In this problem, we are dealing with the coefficient of kinetic friction (bc_k), which relates to objects in motion. For the hockey puck on ice, bc_k is 0.013, indicating that the surface is very smooth. A low value like this signifies minimal resistance to motion, which is why hockey pucks glide easily over ice. The frictional force, which opposes motion, can be calculated using the formula:

• \( F_f = bc_k cdot m cdot g \)

Here, \( g = 9.8 \, \text{m/s}^2 \) is the gravitational acceleration. With such a low friction coefficient, the puck requires only a tiny force to keep moving until eventually slowing down due to friction.

Kinematic equations describe the motion of an object in terms of its initial velocity, final velocity, acceleration, time, and displacement. They are crucial tools for solving mechanics problems in physics. In our scenario, we use the equation:

• \( v_f^2 = v_i^2 + 2ad \)

This kinematic equation is used to find the initial speed \(v_i\) of the puck. Here, \(v_f\) is the final velocity, \(a\) is the acceleration (or in our case, deceleration due to friction), and \(d\) is the distance traveled. The unique part of this problem is that the final velocity \(v_f\) is zero because the puck comes to a stop. This makes the equation easier to solve.

Deceleration is a term for negative acceleration; it indicates that an object is slowing down rather than speeding up. In this exercise, deceleration is caused by the kinetic friction as the puck slides across the ice. We calculate it using the formula:

• \( a = -bc_k cdot g \)

The negative sign indicates that the acceleration is opposing the direction of the puck's motion. Inserting the given values, we determine the deceleration of the puck to be:

• \( a = -0.013 cdot 9.8 \, \text{m/s}^2 \)

This is a very slight deceleration, reinforcing how smoothly and gradually the puck slows down, given the ice's slippery nature.

Calculating the initial speed involves rearranging the kinematic equation to solve for \(v_i\). After setting \(v_f = 0\) in the equation \(v_f^2 = v_i^2 + 2ad\), it simplifies to:

• \( 0 = v_i^2 + 2(-bc_k cdot g)d \)
• \( v_i = \sqrt{2 cdot bc_k cdot g cdot d} \)

Substituting the known values of \(bc_k = 0.013\), \(g = 9.8 \, \text{m/s}^2\), and \(d = 61 \, \text{m}\), we calculate:

• \( v_i = \sqrt{2 cdot 0.013 cdot 9.8 cdot 61} \)
• \( v_i \approx 3.13 \, \text{m/s} \)

This speed is the minimal speed required for the puck to reach the end of the rink before coming to a stop, showcasing the interplay between initial speed, friction, and distance.