A hockey puck slides with an initial speed of 43.2 m/s on a large frozen lake. If the coefficient of kinetic friction between the puck and the ice is 0.036, what is the speed of the puck after 10.4 s?

To determine the speed of the hockey puck after 10.4 s, we can use the equations of motion along with the concept of kinetic friction.

First, let's determine the acceleration of the puck using the coefficient of kinetic friction. The equation for kinetic friction is:

f_k = μk * N

where f_k is the force of kinetic friction, μk is the coefficient of kinetic friction, and N is the normal force. On a flat surface, such as the ice, the normal force is equal to the weight of the object, which is given by:

N = mg

where m is the mass of the puck and g is the acceleration due to gravity.

Since the problem does not provide the mass of the puck, we can only solve for the acceleration up to this point. Assuming the mass of the puck cancels out when determining the acceleration, we can use the following equation:

f_k = μk * mg

Next, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the product of its mass and acceleration:

f_net = ma

Since the only horizontal force acting on the puck is the force of kinetic friction, we have:

f_k = ma

Setting the above equations equal to each other, we have:

μk * mg = ma

Simplifying the equation by canceling out the mass:

μk * g = a

Now, we can calculate the acceleration of the puck using the given coefficient of kinetic friction:

a = μk * g

Substituting the given values for the coefficient of kinetic friction and the acceleration due to gravity:

a = 0.036 * 9.8 m/s^2

Calculating the acceleration:

a = 0.3528 m/s^2

Now, we can use the equation of motion to find the final velocity of the puck after 10.4 s. The equation is:

v_f = v_i + at

where v_f is the final velocity, v_i is the initial velocity, a is the acceleration, and t is the time.

Substituting the given values:

v_f = 43.2 m/s + (0.3528 m/s^2 * 10.4 s)

Calculating the final velocity:

v_f = 43.2 m/s + 3.66912 m/s

v_f = 46.86912 m/s

Therefore, the speed of the puck after 10.4 s is approximately 46.9 m/s.