In attempting to pass the puck to a teammate, a hockey player gives it an initial speed of 2.0m/s, but because of the kinetic friction between the puck and the ice, the puck travels only half the distance between the players before coming to rest. Assuming that the ice's friction is the same everywhere, what minimum initial speed did the player need to give the puck in order for it to reach the teammate?
The puck will need to do twice as much work against friction in order to travel twice as far. That means the initial kinetic energy, (1/2)MV^2, will have to be twice as high.
Since M will be the same, V must increase by a factor sqrt 2 = 1.414
The new V = 2.828 m/s
To find the minimum initial speed that the player needs to give the puck in order for it to reach the teammate, we need to consider the forces acting on the puck.
First, let's analyze the forces acting on the puck when it is in motion:
1. The initial speed provided by the player gives the puck an initial momentum in the forward direction.
2. The kinetic friction between the puck and the ice opposes the motion of the puck, causing it to slow down.
3. There may be other negligible forces acting on the puck, such as air resistance, but for simplicity, let's ignore them in this case.
Now, let's break down the problem and calculate the minimum initial speed required:
1. Determine the distance traveled by the puck before coming to rest:
- The puck travels only half the distance between the players before coming to rest. Let's call this distance "d."
2. Determine the acceleration due to kinetic friction:
- The force of kinetic friction opposes the motion of the puck and causes it to slow down.
- The acceleration due to friction is given by the equation: a = F_friction / m
- Since the puck comes to rest, the final velocity is 0, so we can use the equation: v^2 = u^2 + 2ad
- Rearranging the equation, we get: a = -u^2 / (2d), where "u" is the initial speed.
3. Determine the force of kinetic friction:
- The force of kinetic friction is given by the equation: F_friction = μ * N
- The normal force (N) is equal to the weight of the puck since it is on a horizontal surface. Let's call the weight "W."
- The coefficient of kinetic friction (μ) is a property of the material in contact. Let's call this coefficient "μ_k."
4. Calculate the minimum initial speed required:
- Since the force of friction is in the opposite direction of the initial speed, it will be negative.
- Substitute the expressions for force of friction and acceleration into the equation derived in step 2.
- Solve for the initial speed "u."
By following these steps, you can calculate the minimum initial speed required for the puck to reach the teammate.