Problem 22 - College Physics

Question

A test driver at Incredible Motors, Inc., is testing a new model car having a speedometer calibrated to read \(\mathrm{m} / \mathrm{s}\) rather than \(\mathrm{mi} / \mathrm{h}\). The following series of speedometer readings was obtained during a test run: $$\begin{array}{l|lllllllll}\text { Time (s) } & 0 & 2 & 4 & 6 & 8 & 10 & 12 & 14 & 16 \\\\\hline \text { Velocity (m/s) } & 0 & 0 & 2 & 5 & 10 & 15 & 20 & 22 & 22\end{array}$$ (a) Compute the average acceleration during each 2 s interval. Is the acceleration constant? Is it constant during any part of the test run? (b) Make a velocity-time graph of the data shown, using scales of \(1 \mathrm{~cm}=1\) s horizontally and \(1 \mathrm{~cm}=2 \mathrm{~m} / \mathrm{s}\) vertically. Draw a smooth curve through the plotted points. By measuring the slope of your curve, find the magnitude of the instantaneous acceleration at times \(t=9 \mathrm{~s}, 13 \mathrm{~s},\) and \(15 \mathrm{~s}\)

Step-by-Step Solution

Answer

Average acceleration varies; it's not constant. Use graph slope for instantaneous acceleration.

Step 1: Understand the Problem

We are given a series of velocity readings at specific times and are asked to find the average acceleration during specific intervals and create a velocity-time graph. We need to determine whether acceleration is constant and measure instantaneous acceleration at certain times by analyzing the graph slope.

Step 2: Calculate Average Acceleration

Average acceleration over a time interval is calculated as \( \frac{v_f - v_i}{t_f - t_i} \), where \(v_f\) and \(v_i\) are the final and initial velocities, and \(t_f\) and \(t_i\) are the final and initial times. Compute this for each 2-second interval between the times provided. Ensure that you handle each interval separately from 0 to 16 seconds.

Step 3: Evaluate Constant Acceleration

Check if the average acceleration is the same across multiple intervals. Acceleration is constant if it remains the same across all intervals. Recognize intervals where it stays the same if any exist.

Step 4: Construct Velocity-Time Graph

Create a velocity-time graph using the given scales (1 cm for each second on the x-axis and 2 m/s on the y-axis). Plot the velocity points (time, velocity) from the table and draw a smooth curve connecting these points.

Step 5: Analyze the Graph for Instantaneous Acceleration

To find the instantaneous acceleration at specific times (9s, 13s, and 15s), measure the slope of the tangent at these points on the graph. The slope (change in velocity over change in time) at any instant gives the instantaneous acceleration.

Key Concepts

Velocity-Time Graph

A velocity-time graph is an incredible tool for visualizing an object's motion over a period. In such graphs, time is plotted on the x-axis, while velocity is plotted on the y-axis. By interpreting this graph:

• You can quickly identify how the velocity of an object changes over time.
• The slope of the graph at any point is instrumental in determining acceleration.
• Straight lines suggest constant acceleration or deceleration, while curves indicate varying acceleration.

To create a velocity-time graph from your exercise data:
1 cm on the horizontal axis represents each second, and 1 cm on the vertical axis equates to 2 m/s. Plot each time (seconds) against its corresponding velocity reading. Join these points smoothly, focusing on the overall curve's trend to understand the car's behavior better.

Instantaneous Acceleration

Instantaneous acceleration refers to how quickly the velocity of an object changes at a specific instant. Unlike average acceleration, which is calculated over an interval, instantaneous acceleration provides a snapshot of how quickly the object's speed changes at any single point in time.
To find the instantaneous acceleration:

• Examine your velocity-time graph.
• Identify the tangent line at the time you are interested in (9 seconds, 13 seconds, or 15 seconds).
• Calculate the slope of this tangent line, as this will represent the rate of velocity change at that instant.

Since the slope of a tangent gives instantaneous acceleration, a steep slope indicates higher acceleration, while a flatter slope suggests lower acceleration.

Physics Problem Solving

Solving physics problems, like the one involving velocity and acceleration, requires systematic thinking and a structured approach. Here's a simplified process that can lead you to the solution:

• Understand the problem: Carefully read the question to grasp what is being asked and identify the given information.
• Identify relationships: Use known equations, like acceleration \( a = \frac{\Delta v}{\Delta t} \), to link required outcomes with given data.
• Model visually: Sketching graphs like the velocity-time graph helps visualize changes and trends in motion.
• Check consistency: Verify consistency in your results, such as noticing constant acceleration if the graph's slopes remain the same across intervals.

This structured approach makes complex problems more manageable and enhances solution accuracy.

Kinematics

Kinematics, part of classical mechanics, deals with the motion of objects without worrying about the forces behind them. In our problem context:

• It focuses on parameters like displacement, velocity, and acceleration.
• Average acceleration, for example, is crucial for understanding how quickly an object picks up or drops speed over a given time period.
• The instantaneous acceleration highlights how the velocity is changing at a precise moment, offering deeper insights into the object's dynamics.

By using these kinematic concepts, such as plotting data and analyzing graph slopes in the velocity-time graph, you can get a clearer picture of the vehicle’s dynamic motion during the test run.