Problem 50 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The distance \(s\) (in feet) covered by a car traveling along a straight road is related to its initial speed \(u\) (in \(\mathrm{ft} / \mathrm{sec})\), its final speed \(v\) (in \(\mathrm{ft} / \mathrm{sec})\), and its (constant) acceleration \(a\) (in \(\mathrm{ft} / \mathrm{sec}^{2}\) ) by the equation \(v^{2}=u^{2}+2 a s\). a. Solve the equation for \(a\) in terms of the other variables. b. A car starting from rest and accelerating at a constant rate reaches a speed of \(88 \mathrm{ft} / \mathrm{sec}\) after traveling \(\frac{1}{4}\) mile \((1320 \mathrm{ft})\). What is its acceleration?

Step-by-Step Solution

Answer

The car's acceleration is approximately \(2.93 \, \text{ft}/\text{sec}^2\).

Step 1: Write down the equation

The equation given is \(v^2 = u^2 + 2as\).

Step 2: Isolate acceleration, 'a'

Subtract \(u^2\) from both sides and then divide by \(2s\), to isolate acceleration, 'a'. The equation becomes: \(a = \frac{v^2 - u^2}{2s}\). #b. A car starting from rest and accelerating at a constant rate reaches a speed of 88 ft/sec after traveling 1/4 mile (1320 ft). What is its acceleration?#

Step 1: Identify given values

The initial speed (u) is 0 because the car starts from rest. The final speed (v) is 88 ft/sec, and the distance covered (s) is 1320 ft.

Step 2: Plug the values into the equation

Now, substitute the values of u, v, and s into the equation for acceleration: \(a = \frac{(88)^2 - (0)^2}{2(1320)}\).

Step 3: Simplify the equation

Compute the acceleration by simplifying the equation: \(a = \frac{7744}{2640}\).

Step 4: Calculate acceleration

Divide the numerator by the denominator to find the acceleration: \(a \approx 2.93 \, \text{ft}/\text{sec}^2\). The car's acceleration is approximately 2.93 ft/sec².

Key Concepts

Kinematics

Kinematics is a branch of physics that focuses on the motion of objects. It describes movement in terms of displacement, velocity, and acceleration, without considering the forces causing the motion. In our example problem, we deal with a car traveling along a straight path. When approaching kinematics problems, it's crucial to understand the relationship between different quantities.

• Displacement ( \(s\) ): This is the distance an object moves. In our exercise, the car travels a known distance of 1320 feet.
• Speed (\(u\) and \(v\)): The car's movement begins at an initial speed (\(u\)) and reaches a final speed (\(v\)).
• Acceleration ( \(a\) ): This measures the change in velocity over time, given as a constant in the equation.

By using these concepts, we can determine the acceleration needed to reach a certain speed over a given distance.

Acceleration Calculation

Acceleration is a key concept in kinematics. It is the rate at which the velocity of an object changes. In the provided exercise, we calculate the acceleration needed for a car to reach a certain speed from rest over a specific distance.To find acceleration ( \(a\) ), we rearrange the kinematic equation: \[ v^2 = u^2 + 2as \] Here, \(v\) is the final velocity, \(u\) is the initial velocity (0 in this case), and \(s\) is the displacement. By isolating \(a\), we derive the formula: \[ a = \frac{v^2 - u^2}{2s} \] Now, given that the car starts from rest, initial velocity \(u\) is 0, and we plug in \(v = 88 \, \text{ft/sec}\) and \(s = 1320 \, \text{ft}\).After performing the calculations, we find:

• Calculate \(v^2\): 88 multiplied by 88 equals 7744.
• Calculate \(2s\): 2 times 1320 equals 2640.
• Divide 7744 by 2640 to find \(a\): yields approximately 2.93 \, \text{ft/sec}^2.

This calculated value represents the constant acceleration necessary for the car to achieve the given final speed.

Problem-Solving Steps

When tackling physics problems, understanding the process is as vital as the solution itself. Let's breakdown the problem-solving steps for the exercise.

• Understand the Situation: First, clearly define what is happening in the problem. Here, a car accelerates from rest to a final speed over a specific distance.
• Write Down Given Values: List all known quantities, such as initial and final speeds, as well as the distance of travel. Use these to help identify the unknowns to solve for.
• Use the Correct Formula: Select the appropriate kinematic equation. In this case, the formula relates speed, distance, and acceleration.
• Isolate the Desired Variable: Manipulate the equation to solve for the unknown variable, which is acceleration \(a\).
• Substitute and Simplify: Input the known values into the equation and simplify it to calculate the result. Make sure to double-check calculations for accuracy.
• Verify Your Answer: Finally, re-evaluate the problem to verify that the answer makes sense in context.

By following these structured steps, you can systematically tackle similar kinematic problems and build a solid understanding of physics principles.