Problem 3 - Active Calculus

Question

Suppose that an accelerating car goes from 0 mph to 61.4 mph in five seconds. Its velocity is given in the following table, converted from miles per hour to feet per second, so that all time measurements are in seconds. (Note: \(1 \mathrm{mph}\) is \(22 / 15 \mathrm{ft} / \mathrm{sec} .)\) Find the average acceleration of the car over each of the first two seconds. $$\begin{array}{|l|l|l|l|l|l|l|} \hline t(\mathrm{~s}) & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline v(t)(\mathrm{ft} / \mathrm{s}) & 0.00 & 30.68 & 53.18 & 69.55 & 81.82 & 90.00 \\ \hline \end{array}$$ average acceleration over the first second \(=\square\) help (units) average acceleration over the second second =\(\square\) help (units)

Step-by-Step Solution

Answer

Average acceleration over first second is 30.68 ft/s\textsuperscript{2}, over the second second is 22.5 ft/s\textsuperscript{2}.

Step 1: Understand the Problem

The goal is to find the average acceleration of a car over specific time intervals. The velocity data is given at different time points.

Step 2: Convert Given Information

Note that the velocity values are already converted from miles per hour to feet per second using the provided conversion factor (1 mph = 22/15 ft/s). No further conversion is needed.

Step 3: Find Velocity Change for First Second

We need to determine the velocity change over the first second. From the table, at t=0 seconds, the velocity is 0 ft/s, and at t=1 second, the velocity is 30.68 ft/s. So, the change in velocity is 30.68 ft/s - 0 ft/s = 30.68 ft/s.

Step 4: Calculate Average Acceleration for First Second

Average acceleration is given by the formula \(a = \frac{Δv}{Δt}\). Here, the change in time (Δt) is 1 second. So, average acceleration over the first second is \[a = \frac{30.68 \text{ ft/s}}{1 \text{ s}} = 30.68 \text{ ft/s}^2.\]

Step 5: Find Velocity Change for Second Second

For the second second (from t=1 to t=2 seconds), the velocity at t=1 second is 30.68 ft/s, and at t=2 seconds, the velocity is 53.18 ft/s. So, the change in velocity is 53.18 ft/s - 30.68 ft/s = 22.5 ft/s.

Step 6: Calculate Average Acceleration for Second Second

Using the same formula \(a = \frac{Δv}{Δt}\), the change in time (Δt) is 1 second here as well. So, the average acceleration over the second second is \[a = \frac{22.5 \text{ ft/s}}{1 \text{ s}} = 22.5 \text{ ft/s}^2.\]

Key Concepts

Velocity Change

Velocity change refers to the difference in velocity over a specific period. When dealing with acceleration, understanding how the velocity of an object varies over time is crucial. In our exercise, we dealt with a car accelerating over seconds. Velocity is provided at different points:

• At 0 seconds: 0 ft/s
• At 1 second: 30.68 ft/s
• At 2 seconds: 53.18 ft/s

For the first second, the velocity changed from 0 ft/s to 30.68 ft/s, giving a velocity change of 30.68 ft/s.

For the second second, it changed from 30.68 ft/s to 53.18 ft/s, resulting in a velocity change of 22.5 ft/s.

Understanding how to calculate the change in velocity is essential in determining acceleration.

Time Intervals

The concept of time intervals is closely linked to determining acceleration. Time intervals refer to the periods over which we are measuring changes, such as changes in velocity.

In the context of our example, we looked at two main time intervals:

• From 0 to 1 second (first second)
• From 1 to 2 seconds (second second)

Each time interval represents a one-second period during which we measured the car's velocity. By focusing on these intervals, we can correctly assess the changes in velocity specific to each period.

Understanding these time intervals ensures we correctly apply the acceleration formula.

Acceleration Formula

Acceleration is a measure of how quickly velocity changes over time. To find average acceleration, we use the formula:

\( a = \frac{Δv}{Δt} \)

Where:
\(Δv \) = change in velocity
\(Δt \) = change in time

In our example, we had two intervals:

First Second:
\(Δv = \) 30.68 ft/s (velocity change)
\(Δt = \) 1 second (time interval)
\( a = \frac{30.68 \text{ ft/s}}{1 \text{ s}} = 30.68 \text{ ft/s}^2 \)

Second Second:
\(Δv = \) 22.5 ft/s
\(Δt = \) 1 second
\( a = \frac{22.5 \text{ ft/s}}{1 \text{ s}} = 22.5 \text{ ft/s}^2 \)

Applying the acceleration formula in each time interval allowed us to find the car's average acceleration during those intervals. Always ensure to identify the correct velocity and time changes before using this formula.