In attempting to pass the puck to a teammate, a hockey player gives it an initial speed of 2.82 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 2.82 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 apply the concept of work and kinetic friction. The work done by the applied force is equal to the change in kinetic energy of the puck.

First, let's calculate the work done by the applied force. The work done is given by the equation:

Work = Force x Distance

In this case, the force is the applied force, and the distance is half the distance between the players. Therefore, the work done by the applied force is:

Work = Applied force x (Distance/2)

Next, let's determine the work done by the force of kinetic friction. The work done by friction is equal to the negative change in kinetic energy of the puck. The change in kinetic energy is given by the equation:

Change in kinetic energy = (1/2) x (mass) x (final velocity)^2 - (1/2) x (mass) x (initial velocity)^2

Since the puck comes to a halt, the final velocity is zero. Therefore, the equation simplifies to:

Change in kinetic energy = -(1/2) x (mass) x (initial velocity)^2

According to the problem, the same force of kinetic friction acts on the puck everywhere between the two players. This means that the work done by friction is equal to the force of friction multiplied by the distance traveled.

Now, the work done by the force of kinetic friction is given by:

Work = Force of kinetic friction x Distance

Since the puck comes to a halt, the work done by the force of kinetic friction is equal to the negative change in kinetic energy. Therefore:

-(1/2) x (mass) x (initial velocity)^2 = Force of kinetic friction x Distance

We can simplify this equation by substituting the formula for work done by the applied force:

Force x (Distance/2) = Force of kinetic friction x Distance

Now, let's solve for the minimum initial speed required for the puck to reach the teammate. Rearranging the equation, we get:

Initial velocity^2 = (2 x Force of kinetic friction x Distance) / mass

Taking the square root of both sides, we get:

Initial velocity = sqrt((2 x Force of kinetic friction x Distance) / mass)

Therefore, the minimum initial speed that the puck should have been given to reach the teammate can be calculated using the formula above.

Note: In order to fully solve the problem, we would need specific values for the force of kinetic friction, the distance between the players, and the mass of the puck.