In attempting to pass the puck to a teammate, a hockey player gives it an initial speed of 1.1 m/s. However, this speed is inadequate to compensate for the kinetic friction between the puck and the ice. As a result, the puck travels only one-half the distance between the players before sliding to a halt. What minimum initial speed should the puck have been given so that it reached the teammate, assuming that the same force of kinetic friction acted on the puck everywhere between the two players?

Question

In attempting to pass the puck to a teammate, a hockey player gives it an initial speed of 1.1 m/s. However, this speed is inadequate to compensate for the kinetic friction between the puck and the ice. As a result, the puck travels only one-half the distance between the players before sliding to a halt. What minimum initial speed should the puck have been given so that it reached the teammate, assuming that the same force of kinetic friction acted on the puck everywhere between the two players?

Answer 1

To solve this problem, we need to use the concept of work. The work done by the initial speed of the player on the puck should be equal to the work done by friction in order for the puck to reach the teammate.

The work done by the initial speed on the puck is given by the formula:

Work = (1/2) * mass * (initial speed)^2

The work done by friction is equal to the force of friction multiplied by the distance traveled. Since the work done by friction is equal to the work done by the initial speed, we have:

Force of friction * distance = (1/2) * mass * (initial speed)^2

Now, let's consider the given information. The puck travels only one-half the distance between the players, which means the distance traveled is half the total distance. Let's call the total distance between the players "x". Therefore, the distance traveled is (1/2) * x.

Now we substitute this into the equation:

Force of friction * (1/2) * x = (1/2) * mass * (initial speed)^2

We are asked to find the minimum initial speed, so we want to solve for (initial speed). To remove unnecessary variables, let's assume that the mass of the puck and the force of friction are constants. We can then denote them as "m" and "f" respectively.

Therefore, our equation becomes:

f * (1/2) * x = (1/2) * m * (initial speed)^2

Next, we can simplify the equation:

f * x = m * (initial speed)^2

Now, we solve for the initial speed:

(initial speed)^2 = (f * x) / m

initial speed = sqrt((f * x) / m)

Finally, plug in the given values to find the minimum initial speed required to reach the teammate.