Problem 85 - College Physics

Question

You are traveling north and make a \(90^{\circ}\) right-hand turn east on a flat road while driving a car that has a total weight of 3600 lb. Before the turn, the car was traveling at \(40 \mathrm{mi} / \mathrm{h},\) and after the turn is completed you have slowed to \(30 \mathrm{mi} / \mathrm{h}\). If the turn took \(4.25 \mathrm{~s}\) to complete, determine the following: (a) the car's change in kinetic energy, (b) the car's change in momentum (including direction), and (c) the average net force exerted on the car during the turn (including direction).

Step-by-Step Solution

Answer

Change in KE = -84,179 ft·lb; Change in momentum = 8,177.5 slug·ft/s (NE); Net force = 1,922.4 lbs (NE).

Step 1: Calculate Initial and Final Velocities

Convert the initial and final speeds from miles per hour to feet per second:1 mile = 5280 feet and 1 hour = 3600 seconds.The initial speed is 40 mi/h:\[ v_i = 40 \times \frac{5280}{3600} = \frac{40 \times 5280}{3600} \approx 58.67 \text{ ft/s} \]The final speed is 30 mi/h:\[ v_f = 30 \times \frac{5280}{3600} = \frac{30 \times 5280}{3600} \approx 44.00 \text{ ft/s} \]

Step 2: Compute Initial and Final Kinetic Energy

The kinetic energy \( KE \) is given by the formula:\[ KE = \frac{1}{2}mv^2 \]First, convert the car's weight (3600 lb) to mass:1 pound-force (lb) = 1/32.2 slug (slugs are the unit of mass in the imperial system where gravity is 32.2 ft/s²).\[ m = \frac{3600}{32.2} \approx 111.80 \text{ slugs} \]Initial kinetic energy:\[ KE_i = \frac{1}{2} \times 111.80 \times (58.67)^2 \approx 192,375 \text{ ft} ext{·} ext{lb} \]Final kinetic energy:\[ KE_f = \frac{1}{2} \times 111.80 \times (44.00)^2 \approx 108,196 \text{ ft} ext{·} ext{lb} \]

Step 3: Calculate Change in Kinetic Energy

The change in kinetic energy is:\[ \Delta KE = KE_f - KE_i \]\[ \Delta KE = 108,196 - 192,375 = -84,179 \text{ ft} ext{·} ext{lb} \]

Step 4: Calculate Change in Momentum

Momentum \( p \) is the product of mass and velocity. First compute the initial and final momentum:Initial momentum:\[ p_i = m \cdot v_i = 111.80 \cdot 58.67 = 6,558.7 \text{ slug} ext{·} ext{ft/s} \]Final momentum:\[ p_f = m \cdot v_f = 111.80 \cdot 44.00 = 4,919.2 \text{ slug} ext{·} ext{ft/s} \]Since the change in direction (north to east) also occurs, use vector subtraction to find the change in momentum:The change in momentum magnitude:\[ \Delta p = \sqrt{(p_f)^2 + (p_i)^2} \approx \sqrt{(4,919.2)^2 + (6,558.7)^2} \approx 8,177.5 \text{ slug} ext{·} ext{ft/s} \]Direction: Since the initial direction is north and final is east, the resultant change in momentum is toward northeast at a 45° angle.

Step 5: Calculate the Average Net Force

The average net force \( F_{net} \) can be found using the formula relating force, change in momentum, and time:\[ F_{net} = \frac{\Delta p}{\Delta t} \]Time \( \Delta t = 4.25 \) s.\[ F_{net} = \frac{8,177.5}{4.25} \approx 1,922.4 \text{ lbs (in NE direction)} \]

Key Concepts

Momentum

Momentum is a key concept in physics that helps us understand how an object's motion is influenced by its mass and velocity. It is defined as the product of mass (\( m \)) and velocity (\( v \)), and is expressed as \( p = m imes v \).

In the context of our exercise, we start by calculating the initial and final momentum of a car undergoing a change in direction. The car moves from north to east, which means we need to consider both the magnitude and direction of its momentum. The initial momentum was calculated as \( 6,558.7 \) slug·ft/s moving north and the final momentum as \( 4,919.2 \) slug·ft/s moving east.

To find the change in momentum (\( \Delta p \)), we perform a vector subtraction due to the 90° change in direction. This gives us a magnitude of approximately \( 8,177.5 \) slug·ft/s. Since the movement shifts direction from north to east, this change in momentum is directed northeast, forming a 45° angle with the original direction of travel.

Average Net Force

Average net force is essential for understanding how the velocity and direction of an object change over time when subjected to various forces. It is calculated based on the change in momentum over a certain period.

In this exercise, the turn the car makes involves a change in momentum over \( 4.25 \) seconds. Using the formula:

• \( F_{net} = \frac{\Delta p}{\Delta t} \)

we substitute the values to find the average net force exerted on the car. With \( \Delta p = 8,177.5 \) slug·ft/s and \( \Delta t = 4.25 \) s, the average net force is calculated as \( 1,922.4 \) lbs.

This force is also acting in the northeast direction due to the combined influence of the change in direction and velocity. Understanding average net force helps us grasp how forces interact in real-world situations, such as a car maneuvering a turn.

Physics Problem Solving

Physics problem-solving involves several critical steps to ensure solutions are accurate and meaningful. A systematic approach can greatly aid in tackling complex problems, as illustrated in the car's turning example.

Here's a simple framework you might follow:

• **Identify**: Clearly define what is being asked. In the exercise, we needed to determine changes in kinetic energy, momentum, and average net force.
• **Convert**: Ensure all units are compatible. Convert speeds from miles per hour to feet per second.
• **Calculate**: Use relevant formulas. Calculate kinetic energy, momentum, and force sequentially.
• **Interpret**: Understand the direction of vectors. Determine how changes in parameters like direction affect the results.

By following these steps, you can effectively solve physics problems, ensuring adherence to fundamental principles and avoidance of common pitfalls. Mastering this approach will serve you well in both academic and real-world physics applications.