Problem 16
Question
The fastest drag racers can reach a speed of \(330 \mathrm{mi} /\) hr over a quarter-mile strip in 4.45 seconds (from a standing start). Complete the following sentence about such a drag racer: At some point during the race, the maximum acceleration of the drag racer is at least ___________ \(\mathrm{mi} / \mathrm{hr} / \mathrm{s}\)
Step-by-Step Solution
Verified Answer
Answer: At some point during the race, the maximum acceleration of the drag racer is at least \(\frac{330}{3600 \cdot 4.45}\: \mathrm{mi} / \mathrm{s^2}\).
1Step 1: Convert given speed to \(\mathrm{mi}/\mathrm{s}\)
We are given the final speed as \(330\mathrm{mi}/\mathrm{hr}\). To convert this to \(\mathrm{mi}/\mathrm{s}\), we will divide by 3600, the number of seconds in an hour:
$$
\frac{330\: \mathrm{mi}}{\mathrm{hr}} \cdot \frac{1\: \mathrm{hr}}{3600\: \mathrm{s}} = \frac{330}{3600}\: \mathrm{mi}/\mathrm{s}
$$
2Step 2: Calculate the average acceleration
The average acceleration during the race can be calculated using the formula:
$$
\mathrm{average\:acceleration} =\frac{\text{final\:speed} - \text{initial\:speed}}{\text{time\:taken}}
$$
Since the drag racer starts from rest, the initial speed is 0. The time taken is 4.45 seconds. We can use the value of the final speed in \(\mathrm{mi}/\mathrm{s}\) calculated in Step 1. Therefore, the average acceleration will be:
$$
\mathrm{average\:acceleration} = \frac{\frac{330}{3600}\: \mathrm{mi}/\mathrm{s} - 0}{4.45\: \mathrm{s}}
$$
3Step 3: Simplify the average acceleration
Simplifying the average acceleration expression, we get:
$$
\mathrm{average\:acceleration} = \frac{330}{3600 \cdot 4.45}\: \mathrm{mi}/\mathrm{s^2}
$$
4Step 4: Convince ourselves that maximum acceleration must be at least the average acceleration
Since the drag racer starts from rest and reaches its final speed within 4.45 seconds, it must have accelerated throughout the race. At some point during the race, the drag racer's acceleration is highest, which is the maximum acceleration. For the racer to reach the final speed in the given time, the maximum acceleration must be at least the average acceleration.
5Step 5: Complete the sentence
From Steps 3 and 4, we have shown that the maximum acceleration of the drag racer at some point during the race is at least \(\frac{330}{3600 \cdot 4.45}\: \mathrm{mi}/\mathrm{s^2}\). To complete the sentence:
"At some point during the race, the maximum acceleration of the drag racer is at least \(\frac{330}{3600 \cdot 4.45}\: \mathrm{mi} / \mathrm{hr} / \mathrm{s}\)."
Key Concepts
Drag RacingAverage AccelerationUnit Conversion
Drag Racing
Drag racing is an exhilarating motorsport where two cars race from a standing start, aiming to cross the finish line first. These races usually occur on a straight, quarter-mile track, allowing racers to push their cars to extreme speeds over short distances.
In every drag race, acceleration is a critical factor. Drivers attempt to reach the maximum speed possible in the shortest time. This involves reaching high speeds, such as the impressive 330 miles per hour mentioned in many professional races, usually within several seconds.
Understanding the dynamics of acceleration helps in appreciating the expertise of the racers. They utilize both engines’ maximum power and precise timing to harness the car's potential. This showcases the fascinating interplay between physics and human skill in the sport.
In every drag race, acceleration is a critical factor. Drivers attempt to reach the maximum speed possible in the shortest time. This involves reaching high speeds, such as the impressive 330 miles per hour mentioned in many professional races, usually within several seconds.
Understanding the dynamics of acceleration helps in appreciating the expertise of the racers. They utilize both engines’ maximum power and precise timing to harness the car's potential. This showcases the fascinating interplay between physics and human skill in the sport.
Average Acceleration
When studying motion, especially in drag racing, average acceleration is a vital concept. It describes how quickly speed changes over a specific period.
Average acceleration can be calculated using the formula: - The final speed minus the initial speed, - Divided by the time taken for this change in speed.
In the example of a drag racer:
This calculation provides an understanding of the overall rate of change in velocity, essential for achieving competitive times in races. It gives an insight into the car's performance throughout the event.
Average acceleration can be calculated using the formula: - The final speed minus the initial speed, - Divided by the time taken for this change in speed.
In the example of a drag racer:
- The initial speed is zero since the racer starts from rest.
- The final speed is the speed of 330 miles per hour reached over 4.45 seconds.
This calculation provides an understanding of the overall rate of change in velocity, essential for achieving competitive times in races. It gives an insight into the car's performance throughout the event.
Unit Conversion
An essential part of solving physics problems involves unit conversion, which ensures consistency in calculations. For example, when dealing with speed and acceleration, using consistent units such as miles per second is crucial.
In drag racing calculations, speeds often need converting from miles per hour to miles per second for practical mathematical application.
By ensuring all measurements are in compatible units, such as converting time from hours to seconds, students can compute formulas correctly. This practice underpins not just drag racing calculations but is a fundamental skill across all areas of physics and engineering.
In summary, mastering unit conversion paves the way for accurate problem-solving.
In drag racing calculations, speeds often need converting from miles per hour to miles per second for practical mathematical application.
- To convert, one might divide by 3600, the number of seconds in an hour, helping in scenarios where timing is key.
By ensuring all measurements are in compatible units, such as converting time from hours to seconds, students can compute formulas correctly. This practice underpins not just drag racing calculations but is a fundamental skill across all areas of physics and engineering.
In summary, mastering unit conversion paves the way for accurate problem-solving.
Other exercises in this chapter
Problem 16
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