Wally and Tammy Play Shuffleboard
Wally and Tammy are playing Shuffleboard. The distance between Wally's shooting area and the center of the 10 point zone is 9.5 m. The coefficient of kinetic friction between the puck (0.1 kg) and the court surface is 0.3.
Question 1: What will be the acceleration of Wally's puck as it slides across the court?
There are three forces acting on the puck as it slides: a gravitational force, a normal force from the court, and a force of friction.
Figure 1: Our Sketch of the Forces Acting on the Puck
Only the frictional force acts in the direction of motion. Therefore, using Newton's Second Law of Motion, we know that it must equal the puck's mass times its acceleration:
\[ F_\text{k} = ma \]
We also know that the frictional force is equal to the coefficient of friction, multiplied by the normal force on the puck:
\[ F_\text{k} = \mu_\text{k}N \]
Therefore, our original expression becomes:
\[ \mu_\text{k}N = ma \]
The normal force opposes and balances the gravitational force on the puck exactly, since it is not moving at all in the vertical direction:
\[ N = -mg \]
\[ \mu_\text{k} \times -mg = ma \]
Note that the mass on both sides cancels. The acceleration of the puck does not depend on its mass:
\[ -\mu_\text{k} \times g = a \]
Finally, we solve for acceleration and substitute for known variables:
\[ a = -\mu_\text{k} \times g \]
\[ a \approx -(0.3) \times 9.8 \, \text{m/s}^2 \]
Arriving at our final value for the puck's acceleration:
\[ a \approx -2.94 \, \text{m/s}^2 \]
Question 2: What would the puck's initial velocity need to be in order for Wally to make a perfect shot?
We utilize the Velocity-Position equation from our Equations of Motion. This relates initial and final velocities to initial and final positions.
\[ {v_\text{f}}^2 = {v_\text{i}}^2 + 2a(x_2 - x_1) \]
Rearranging the equation to solve for initial velocity:
\[ {v_\text{i}} = \sqrt{{v_\text{f}}^2 - 2a(x_2 - x_1) } \]
And substituting for known variables:
\[ v_\text{i} \approx \sqrt{(0)^2 - 2 \times (-2.94 \, \text{m/s}^2) \times (9.5 \, \text{m} - 0 \, \text{m}) } \]
Gives us our value for the puck's initial velocity:
\[ v_\text{i} \approx 7.5 \, \text{m/s} \]