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Chapter 2: Problem 50

The Detonator On a ride called the Detonator at Worlds of Fun in Kansas City, passengers accelerate straight downward from rest to \(45 \mathrm{mi} / \mathrm{h}\) in \(2.2 \mathrm{seconds.}\) What is the average acceleration of the passengers on this ride?

Short Answer

Expert verified
The average acceleration is approximately \(9.14 \text{ m/s}^2\).

Step by step solution

01

Understand the problem

The problem asks us to find the average acceleration of passengers on a ride as they accelerate from rest to a specific speed over a certain period of time.
02

Convert units, if necessary

The given speed is in miles per hour, but acceleration is typically measured in meters per second squared. First, convert \(45 \text{ mi/h}\) to meters per second using the conversion factor: \(1 \text{ mi/h} = 0.44704 \text{ m/s}\). So, \(45 \text{ mi/h} \approx 20.1168 \text{ m/s}\).
03

Apply the formula for average acceleration

The formula for average acceleration \(a\) is \(a = \frac{\Delta v}{\Delta t}\), where \(\Delta v\) is the change in velocity and \(\Delta t\) is the change in time. Here, \(\Delta v = 20.1168 \text{ m/s} - 0\), and \(\Delta t = 2.2 \text{ s}\).
04

Calculate the average acceleration

Substitute the known values into the formula: \(a = \frac{20.1168 \text{ m/s}}{2.2 \text{ s}}\). Upon solving, we find \(a \approx 9.144 \text{ m/s}^2\).
05

Confirm units are correct

Ensure that the final answer has appropriate units for acceleration, which are meters per second squared (\(\text{ m/s}^2\)).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Unit Conversion
When dealing with physical quantities, it's crucial to ensure that they are in the correct units before carrying out calculations. In the case of acceleration, which is typically measured in meters per second squared (\( ext{m/s}^2\)), we often need to convert other units into this standard form.
For example, speed given in miles per hour (mi/h) should be converted to meters per second (m/s) because these units align with the International System of Units (SI) used in physics.
  • To convert from mi/h to m/s, we use the conversion factor: \(1 \text{ mi/h} = 0.44704 \text{ m/s}\).
  • This means you multiply the speed value in mi/h by 0.44704 to convert it to m/s.
Understanding and applying unit conversion is a foundational skill in physics, as it ensures consistency and accuracy in computations.
Velocity Change in Motion
Velocity change is a key concept when analyzing motion, as it indicates how fast an object's speed is increasing or decreasing over time. In the context of the Detonator ride, passengers start from rest and reach their maximum speed in a set duration.
When mentioning change in velocity, \(\Delta v\), it specifically refers to the difference between the final velocity and the initial velocity.
  • Final velocity (\(v_f\)) is what the object reaches at the end of the time interval.
  • Initial velocity (\(v_i\)) is the velocity the object starts with. In many problems, like this one, it might be zero if starting from rest.
Understanding how velocity changes over time helps us calculate acceleration and analyze motion.
Applying the Acceleration Formula
The acceleration formula is a powerful tool in physics for determining how quickly an object is changing its velocity over time. It is represented by:\[a = \frac{\Delta v}{\Delta t}\]Here, \(a\) represents average acceleration, \(\Delta v\) is the change in velocity, and \(\Delta t\) is the change in time.
When calculating average acceleration, steps include:
  • Determine \(\Delta v\), which is the end velocity minus the start velocity.
  • Identify \(\Delta t\), the time period over which the change occurs.
  • Substitute these values into the equation to find \(a\).
The resultant value will be in meters per second squared (\( ext{m/s}^2\)), provided the velocity was measured in meters per second and time in seconds. This process allows us to quantify how quickly passengers on an amusement ride accelerate, offering insights into the ride's dynamics.

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Most popular questions from this chapter

Joseph DeLoach of the United States set an Olympic record in 1988 for the 200 -meter dash with a time of 19.75 seconds. What was his average speed? Give your answer in meters per second and miles per hour.

You drive in a straight line at \(20.0 \mathrm{m} / \mathrm{s}\) for \(10.0 \mathrm{minutes}\), then at \(30.0 \mathrm{m} / \mathrm{s}\) for another \(10.0 \mathrm{minutes}\). (a) Is your average speed 25.0 \(\mathrm{m} / \mathrm{s},\) more than \(25.0 \mathrm{m} / \mathrm{s},\) or less than \(25.0 \mathrm{m} / \mathrm{s}\) ? Explain. (b) Verify your answer to part (a) by calculating the average speed.

A boat is cruising in a straight line at a constant speed of \(2.6 \mathrm{m} / \mathrm{s}\) when it is shifted into neutral. After coasting \(12 \mathrm{m}\) the engine is engaged again, and the boat resumes cruising at the reduced constant speed of \(1.6 \mathrm{m} / \mathrm{s}\). Assuming constant acceleration while coasting. (a) how long did it take for the boat to coast the \(12 \mathrm{m} ?\) (b) What was the boat's acceleration while it was coasting? \((c)\) When the boat had coasted for \(6.0 \mathrm{m},\) was its speed \(2.1 \mathrm{m} / \mathrm{s},\) more than \(2.1 \mathrm{m} / \mathrm{s},\) or less than \(2.1 \mathrm{m} / \mathrm{s}\) ? Explain.

A doctor, preparing to give a patient an injection, squirts a small amount of liquid straight upward from a syringe. If the liquid emerges with a speed of \(1.5 \mathrm{m} / \mathrm{s},\) (a) how long does it take for it to return to the level of the syringe? (b) What is the maximum height of the liquid above the syringe?

Assume that the brakes in your car create a constant deceleration of \(4.2 \mathrm{m} / \mathrm{s}^{2}\) regardless of how fast you are driving. If you double your driving speed from \(16 \mathrm{m} / \mathrm{s}\) to \(32 \mathrm{m} / \mathrm{s},\) (a) does the time required to come to a stop increase by a factor of two or a factor of four? Explain. Verify your answer to part (a) by calculating the stopping times for initial speeds of (b) \(16 \mathrm{m} / \mathrm{s}\) and (c) \(32 \mathrm{m} / \mathrm{s}\)
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