Problem 87 - College Physics

Question

A hockey puck impacts a goalie's plastic mask horizontally at \(122 \mathrm{mi} / \mathrm{h}\) and rebounds horizontally off the mask at \(47 \mathrm{mi} / \mathrm{h}\). If the puck has a mass of \(170 \mathrm{~g}\) and it is in contact with the mask for \(25 \mathrm{~ms},\) (a) what is the average force (including direction) that the puck exerts on the mask? (b) Assuming that this average force accelerates the goalie (neglect friction with the ice), with what speed will the goalie move, assuming she was at rest initially and has a total mass of \(85 \mathrm{~kg}\) ?

Step-by-Step Solution

Answer

(a) -227.92 N (opposite direction); (b) Goalie's speed is 0.067 m/s.

Step 1: Convert Units

First, convert the velocities from miles per hour to meters per second. Use the conversion factor 1 mile = 1609.34 meters and 1 hour = 3600 seconds. \[v_{i} = 122 \text{ mi/h} = \frac{122 \times 1609.34}{3600} \approx 54.53 \text{ m/s} \]\[v_{f} = 47 \text{ mi/h} = \frac{47 \times 1609.34}{3600} \approx 21.01 \text{ m/s} \]

Step 2: Calculate Change in Velocity

Determine the change in velocity by subtracting the final velocity from the initial velocity. The initial direction is positive as it was the initial impact direction.\[\Delta v = v_{f} - v_{i} = 21.01 \text{ m/s} - 54.53 \text{ m/s} = -33.52 \text{ m/s} \]

Step 3: Calculate Change in Momentum

Calculate the change in momentum. Using the mass of the puck \(m = 0.170 \text{ kg}\) for calculations:\[\Delta p = m \cdot \Delta v = 0.170 \times (-33.52) = -5.698 \text{ kg} \cdot \text{m/s} \]

Step 4: Calculate Average Force

Use the formula for force based on change in momentum and time to find the average force exerted on the mask.\[F = \frac{\Delta p}{\Delta t} = \frac{-5.698}{0.025} \approx -227.92 \text{ N} \]The negative sign indicates that the force direction is opposite to the initial motion.

Step 5: Calculate Goalies Acceleration

Using the average force and the goalie's mass, calculate the acceleration imparted on the goalie. \[a = \frac{F}{m} = \frac{227.92}{85} \approx 2.68 \text{ m/s}^2 \]

Step 6: Calculate Goalies Speed

Find the goalie's speed after being acted on by the force for the same time duration:\[v_{goalie} = a \cdot \Delta t = 2.68 \cdot 0.025 = 0.067 \text{ m/s} \]

Key Concepts

Change in Velocity

The change in velocity is a critical concept in understanding momentum and impulse. When an object experiences a change in velocity, it means its speed and/or direction has altered. This is due to the presence of an external force. In the exercise, the hockey puck initially moves at a speed of 54.53 meters per second (after converting from miles per hour) and rebounds at 21.01 meters per second.
The change in velocity \[\Delta v = v_f - v_i = 21.01 ext{ m/s} - 54.53 ext{ m/s} = -33.52 ext{ m/s}\]- **Initial velocity**: 54.53 m/s- **Final velocity**: 21.01 m/s- **Change in velocity**: -33.52 m/sThis negative sign indicates the direction of the puck's velocity changed, showing the force from the goalie's mask reversed its motion.

Average Force

Average force plays a fundamental role in determining how a moving object influences another object during a collision. By considering the change in momentum over time, you can calculate the average force exerted in the interaction. In our example, the force exerted by the puck on the goalie's mask can be found using the formula:\[F = \frac{\Delta p}{\Delta t}\]Where:- **\(\Delta p\)** is the change in momentum- **\(\Delta t\)** is the time duration the objects are in contactIn this case, the calculations show:\[F = \frac{-5.698 \text{ kg} \cdot \text{m/s}}{0.025 \text{ s}} \approx -227.92 \text{ N}\]The sign of the force is negative, indicating it acts in the opposite direction of the puck's initial motion. Understanding average force is essential because it helps us determine effects like acceleration generated in the impacted object.

Impulse and Momentum

Impulse and momentum are vital concepts in physics that describe the effects of forces over time on moving objects. Momentum (\( p \)) is the product of mass and velocity (\( p = m \times v \)), and impulse (\( J \)) is the change in momentum caused by a force over a time period.In the exercise, the puck's change in momentum was calculated using its mass (0.170 kg) and the change in velocity (-33.52 m/s):\[\Delta p = m \times \Delta v = 0.170 \times -33.52 = -5.698 \text{ kg} \cdot \text{m/s}\]Impulse is thus:\[J = \Delta p = -5.698 \text{ kg} \cdot \text{m/s}\]- **Initial momentum**: Based on initial velocity- **Final momentum**: Based on final velocity- **Impulse**: Caused by the external force (goalie's mask)Understanding these concepts allows us to predict how different forces affect object motion over time, such as how the goalie's mass and the forces involved translate into her subsequent motion speed.