Problem 90 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A car traveling along a straight road at \(66 \mathrm{ft} / \mathrm{sec}\) accelerated to a speed of \(88 \mathrm{ft} / \mathrm{sec}\) over \(\mathrm{a}\) distance of \(440 \mathrm{ft}\). What was the acceleration of the car, assuming it was constant?

Step-by-Step Solution

Answer

The constant acceleration of the car is approximately \(3.846 \mathrm{ft/s^2}\).

Step 1: Identify given variables and formula

In this problem, we are given: - Initial speed, \(v_i = 66 \mathrm{ft} / \mathrm{sec}\) - Final speed, \(v_f = 88 \mathrm{ft} / \mathrm{sec}\) - Distance, \(d = 440 \mathrm{ft}\) We want to find the acceleration (a), assuming it is constant. We will use the Kinematic Equation: \[v_f^2 = v_i^2 + 2ad\]

Step 2: Substitute known values into the formula

Now, we will substitute the given values into the Kinematic Equation: \[(88 \mathrm{ft/s})^2 = (66 \mathrm{ft/s})^2 + 2a(440\mathrm{ft})\]

Step 3: Solve the equation for 'a'

We will now solve this equation for 'a'. First, square the speeds: \[(7744 \mathrm{ft^2/s^2}) = (4356 \mathrm{ft^2/s^2}) + (880\mathrm{ft})a\] Now we subtract \(4356 \mathrm{ft^2/s^2}\) from both sides of the equation: \[3388 \mathrm{ft^2/s^2} = 880\mathrm{ft}\cdot a\] Next, we divide both sides by \(880\mathrm{ft}\) to isolate 'a': \[a = \frac{3388\mathrm{ft^2/s^2}}{880\mathrm{ft}}\]

Step 4: Calculate the acceleration 'a'

We can now solve for the acceleration: \[a = \frac{3388}{880}\mathrm{ft/s^2} \approx 3.846\mathrm{ft/s^2}\] So, the constant acceleration of the car is approximately \(3.846 \mathrm{ft/s^2}\).

Key Concepts

Constant Acceleration

Constant acceleration describes a scenario where the rate of change of velocity remains the same over time. It means that an object's velocity changes by an equal amount in every equal time period. In physics problems involving motion, assuming constant acceleration simplifies calculations because it allows us to use specific kinematic equations. These equations relate the initial velocity, final velocity, acceleration, and distance.

This assumption is applicable when external forces do not change drastically. For example, in our car problem, the road conditions, and engine power maintain consistent acceleration. We used the kinematic equation \[v_f^2 = v_i^2 + 2ad\] to find the acceleration. In this context: - \(v_f\) is the final velocity,- \(v_i\) is the initial velocity,- \(a\) is the constant acceleration, - \(d\) is the distance traveled.

Knowing any three variables allows you to solve for the fourth. This powerful characteristic is a boon in physics problem-solving whenever constant acceleration can be assumed.

Initial and Final Velocity

Understanding initial and final velocity is key to solving motion problems in physics. Initial velocity refers to the speed and direction an object starts moving with. Final velocity is its speed and direction at the end of a time interval. Both these velocities are vector quantities, meaning they have both magnitude and direction. However, in one-dimensional motion like our example problem, we can often focus solely on magnitude.

In the case of the car,

• The initial velocity \(v_i\) is given as \(66 \mathrm{ft/s}\).
• The final velocity \(v_f\) is \(88 \mathrm{ft/s}\).

These values are instrumental in applying the kinematic equation and finding acceleration. It's essential to clearly distinguish the initial from the final state to avoid confusion. Getting these values wrong can lead to errors in further calculations, as they directly affect the application of formulas. By understanding and clearly identifying these velocities, students can better solve motion-related questions.

Problem-Solving in Physics

Physics is sometimes viewed as a tough subject due to its abstract concepts. But, it follows a systematic way of problem-solving that helps in understanding and tackling these abstract aspects.

The approach involves:

• Clearly identifying and understanding what is being asked.
• Listing all known and unknown variables.
• Selecting the appropriate formula or principle.
• Substituting given data into the equation.
• Solving step by step for the unknown.
• Reviewing the steps and checking if the answer makes sense.

This step-by-step methodology applies not only to kinematic equations but also to other areas in physics. In our problem, each step, from identifying initial and final velocities to using the correct kinematic equation, demonstrates this logical process. Such structured problem-solving makes seemingly complex problems manageable and enhances our ability to understand and approach physics questions confidently.