Problem 16
Question
An accelerating sports car goes from 0 mph to 60 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 mph is \(22 / 15\) ft/sec.) Find the average acceleration of the car over each of the first two seconds. $$\begin{array}{l|c|c|c|c|c|c}\hline \text { Time, } t(\mathrm{sec}) & 0 & 1 & 2 & 3 & 4 & 5 \\\\\hline \text { Velocity, } v(t)(\mathrm{f} t / \mathrm{sec}) & 0 & 30 & 52 & 68 & 80 & 88 \\\\\hline\end{array}$$
Step-by-Step Solution
Verified Answer
30 ft/sec² for the first second, 22 ft/sec² for the second.
1Step 1: Understand Average Acceleration
Average acceleration is defined as the change in velocity over the change in time. Mathematically, this is represented by the formula: \( a_{avg} = \frac{\Delta v}{\Delta t} \), where \( \Delta v \) is the change in velocity and \( \Delta t \) is the change in time.
2Step 2: Calculate Change in Velocity for 0 to 1 Second
At 0 seconds, the velocity is 0 ft/sec, and at 1 second, it is 30 ft/sec. The change in velocity \( \Delta v \) from 0 to 1 second is \( 30 - 0 = 30 \) ft/sec.
3Step 3: Calculate Change in Time for 0 to 1 Second
The change in time \( \Delta t \) from 0 to 1 second is simply \( 1 - 0 = 1 \) second.
4Step 4: Calculate Average Acceleration for 0 to 1 Second
Using the formula for average acceleration: \( a_{avg} = \frac{\Delta v}{\Delta t} \), the average acceleration for 0 to 1 second is \( \frac{30}{1} = 30 \) ft/sec².
5Step 5: Calculate Change in Velocity for 1 to 2 Seconds
At 1 second, the velocity is 30 ft/sec, and at 2 seconds, it is 52 ft/sec. The change in velocity \( \Delta v \) from 1 to 2 seconds is \( 52 - 30 = 22 \) ft/sec.
6Step 6: Calculate Change in Time for 1 to 2 Seconds
The change in time \( \Delta t \) from 1 to 2 seconds is \( 2 - 1 = 1 \) second.
7Step 7: Calculate Average Acceleration for 1 to 2 Seconds
Using the formula for average acceleration: \( a_{avg} = \frac{\Delta v}{\Delta t} \), the average acceleration for 1 to 2 seconds is \( \frac{22}{1} = 22 \) ft/sec².
Key Concepts
velocitychange in velocitychange in timecalculus
velocity
Velocity is the rate at which an object changes its position. Simply put, it's how fast something is moving in a particular direction. It's an important concept in physics because it not only tells us the speed but also the direction of the movement.
For example, a car moving forward at 30 feet per second (ft/sec) has a velocity vector pointing forward with a magnitude of 30 ft/sec. In our sports car exercise, the car goes from 0 ft/sec to various velocities specified in the table over time.
Understanding velocity is crucial because it forms the basis for calculating acceleration, which is a change in velocity over time.
For example, a car moving forward at 30 feet per second (ft/sec) has a velocity vector pointing forward with a magnitude of 30 ft/sec. In our sports car exercise, the car goes from 0 ft/sec to various velocities specified in the table over time.
Understanding velocity is crucial because it forms the basis for calculating acceleration, which is a change in velocity over time.
change in velocity
The change in velocity (\( \Delta v \)) is exactly what it sounds like: it's how much an object's velocity changes over a given period. When calculating average acceleration, it's essential to find the difference between the initial and final velocities.
For instance, in our exercise, the car's velocity changes from 0 ft/sec at the start to 30 ft/sec at the 1-second mark. This results in a change in velocity of \( 30 \text{ ft/sec} - 0 \text{ ft/sec} = 30 \text{ ft/sec} \). Similarly, from 1 to 2 seconds, the velocity changes from 30 ft/sec to 52 ft/sec, resulting in a change of \( 22 \text{ ft/sec} \).
Calculating the change in velocity allows us to understand how the object's speed increases or decreases, which is fundamental for understanding acceleration.
For instance, in our exercise, the car's velocity changes from 0 ft/sec at the start to 30 ft/sec at the 1-second mark. This results in a change in velocity of \( 30 \text{ ft/sec} - 0 \text{ ft/sec} = 30 \text{ ft/sec} \). Similarly, from 1 to 2 seconds, the velocity changes from 30 ft/sec to 52 ft/sec, resulting in a change of \( 22 \text{ ft/sec} \).
Calculating the change in velocity allows us to understand how the object's speed increases or decreases, which is fundamental for understanding acceleration.
change in time
Change in time (\( \Delta t \)) refers to the interval over which changes occur. It is a simple yet critical component of the acceleration formula. Typically, in exercises like our car example, time is measured in seconds.
In our scenario, the change in time for the first segment, from time 0 to 1 second, is \( 1 - 0 = 1 \text{ second} \).
The same change in time is true for the next segment from 1 second to 2 seconds, which is also \( 1 \text{ second} \).
Recognizing the change in time helps us apply it to find the average acceleration over different intervals. It's a straightforward subtraction of the time values, giving us the duration needed to calculate acceleration.
In our scenario, the change in time for the first segment, from time 0 to 1 second, is \( 1 - 0 = 1 \text{ second} \).
The same change in time is true for the next segment from 1 second to 2 seconds, which is also \( 1 \text{ second} \).
Recognizing the change in time helps us apply it to find the average acceleration over different intervals. It's a straightforward subtraction of the time values, giving us the duration needed to calculate acceleration.
calculus
Calculus is the branch of mathematics that deals with continuous change. Although the example provided does not require deep calculus, understanding its role can provide insights into topics such as velocity and acceleration.
In calculus, the concept of a derivative helps us understand instantaneous rates of change. While our example covers average acceleration, calculus can be applied to find an instantaneous acceleration—a kind of acceleration at a specific instant—by taking the derivative of velocity with respect to time.
In calculus, the concept of a derivative helps us understand instantaneous rates of change. While our example covers average acceleration, calculus can be applied to find an instantaneous acceleration—a kind of acceleration at a specific instant—by taking the derivative of velocity with respect to time.
- In simple terms, calculus allows us to precisely calculate changes in quantities, not just over intervals like seconds, but at any specific moment.
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