Problem 45 - Calculus an Applied Approach

Question

Velocity and Acceleration The velocity (in feet per second) of an automobile starting from rest is modeled by \(\frac{d s}{d t}=\frac{90 t}{t+10}\) Create a table showing the velocity and acceleration at 10 -second intervals during the first minute of travel. What can you conclude?

Step-by-Step Solution

Answer

From this, we can conclude that the car accelerates less as time progresses and the maximum speed it can reach is 90 feet per second.

Step 1: Calculate Velocity

At every 10 second interval between 0 and 60 seconds, substitute the time \(t\) into the given velocity function \(v(t) = \frac{90 t}{t+10}\), and compute the resulting velocity.

Step 2: Determine Acceleration Function

Differentiate the velocity function \(v(t) = \frac{90 t}{t+10}\) to obtain the acceleration function \(a(t) = \frac{900}{(t+10)^2}\). This can be done using the quotient rule for derivatives.

Step 3: Calculate Acceleration

At every 10 second interval between 0 and 60 seconds, substitute the time \(t\) into the acceleration function \(a(t) = \frac{900}{(t+10)^2}\), and compute the resulting acceleration.

Step 4: Create table

Using the velocity and acceleration values obtained in steps 1 and 3 respectively, generate a table with columns labeled 'Time in seconds', 'Velocity', and 'Acceleration', and rows for each 10 second interval between 0 and 60 seconds.

Step 5: Interpret the Data

Examine the values in the table to discern any patterns or trends in the velocity or acceleration over time. In this case, both the velocity and acceleration are non-negative and decrease as time increases. The velocity tends to a constant indicating a limit to the speed the automobile can reach, while the decreasing acceleration signifies the vehicle is slowing down.

Key Concepts

Quotient Rule

To solve many problems involving derivatives, the quotient rule is essential. When we have a function that is the division of two other functions, like in the case of velocity functions, the quotient rule helps us differentiate it. To apply the quotient rule, we express our function as \(f(t) = \frac{u(t)}{v(t)}\). Here, \(u(t)\) and \(v(t)\) are functions of \(t\). The rule is:

• Take the derivative of the top function \(u(t)\), denoted as \(u'(t)\).
• Take the derivative of the bottom function \(v(t)\), denoted as \(v'(t)\).
• Plug these into the formula: \(f'(t) = \frac{u'(t)v(t) - u(t)v'(t)}{[v(t)]^2}\).

Whenever you have a fraction, the quotient rule is your go-to method for finding the derivative.

Velocity Function

The velocity function shows how the speed of an object changes over time. In this case, our velocity function is \(v(t) = \frac{90t}{t+10}\). This particular function helps to calculate the speed of a car starting from rest, where the variable \(t\) represents time in seconds.

• At \(t = 0\), the car isn't moving, as it just starts from rest;
• The function indicates how the car speeds up as time goes on;
• The denominator increases faster than the numerator over time.

Given this function, we notice that while the velocity increases, its rate of increase becomes slower, indicating a leveling out as time goes by. This is because the car approaches a maximum velocity it can reach.

Acceleration Function

The acceleration function takes the derivative of the velocity function, showing how the change in speed itself is changing over time. Using the quotient rule on our velocity function \(v(t) = \frac{90t}{t+10}\), we find the acceleration function to be \(a(t) = \frac{900}{(t+10)^2}\).

• This means acceleration decreases as time increases;
• Initially, the car has a higher acceleration, which decreases over time;
• Acceleration is found by evaluating \(a(t)\) at different values of \(t\).

Thus, the acceleration function helps us understand how the velocity levels out and approaches a steady speed, despite continually positive values.

Derivatives in Calculus

Derivatives are one of the fundamental building blocks of calculus, crucial for understanding concepts like velocity and acceleration. They measure how a function changes as its input changes. In our case of the car breaking down the velocity and acceleration, derivatives provide a precise mathematical way to comprehend these changing rates.

• The first derivative tells us the rate of change of the function, which for our velocity function reflects the car's speed;
• The second derivative, which is the derivative of the velocity function, becomes the acceleration function;
• Understanding derivatives allows prediction and analysis of physical phenomena like motion.

Utilizing derivatives, we gain insight into the behavior of moving objects, ensuring a deeper grasp of dynamic systems in the realm of physics and beyond.