/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 (a) Suppose that a NASCAR race c... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 15

(a) Suppose that a NASCAR race car is moving to the right with a constant velocity of \(+82 \mathrm{m} / \mathrm{s}\) . What is the average acceleration of the car? (b) Twelve seconds later, the car is halfway around the track and traveling in the opposite direction with the same speed. What is the average acceleration of the car?

Short Answer

Expert verified
(a) \(0 \mathrm{m/s}^2\); (b) \(-13.67 \mathrm{m/s}^2\).

Step by step solution

01

Understanding Average Acceleration

Average acceleration is defined as the change in velocity divided by the time during which the change occurred. It can be calculated using 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.
02

Calculate Change in Velocity for Part (a)

For part (a), since the car is moving with a constant velocity of \(+82 \mathrm{m/s}\), there is no change in velocity.\[ \Delta v = v_f - v_i = 82 \mathrm{m/s} - 82 \mathrm{m/s} = 0 \mathrm{m/s} \]Thus, \( \Delta t \) can be any value, but since \( \Delta v = 0 \), there is no acceleration.
03

Finding Average Acceleration for Part (a)

Using the change in velocity calculated in Step 2:\[ a_{avg} = \frac{0 \mathrm{m/s}}{\Delta t} = 0 \mathrm{m/s}^2 \]Therefore, the average acceleration is \(0 \mathrm{m/s}^2\).
04

Calculate Change in Velocity for Part (b)

For part (b), the initial velocity is \(+82 \mathrm{m/s}\) and the final velocity is \(-82 \mathrm{m/s}\) since the car is traveling in the opposite direction.\[ \Delta v = v_f - v_i = -82 \mathrm{m/s} - (+82 \mathrm{m/s}) = -164 \mathrm{m/s} \]
05

Finding Average Acceleration for Part (b)

Given \( \Delta t = 12 \) seconds:\[ a_{avg} = \frac{-164 \mathrm{m/s}}{12 \text{ s}} = -13.67 \mathrm{m/s}^2 \]So, the average acceleration is \(-13.67 \mathrm{m/s}^2\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Constant Velocity
When a NASCAR race car is traveling at a constant velocity, it is moving at a uniform speed in a straight line. This means there is no change in speed or direction. Consequently, the change in velocity is zero.
Since there is no variation in the car's motion, it does not accelerate. Acceleration occurs only when there is a change in velocity, such as speeding up, slowing down, or changing direction.
  • Constant velocity translates to zero acceleration.
  • Speed remains uniform.
  • No directional change is involved.
This concept helps us understand that the average acceleration for an object with constant velocity is zero. Any change in velocity would mean the object is no longer moving at a constant speed.
Change in Velocity
Change in velocity is a critical factor in understanding acceleration. It measures how the speed and direction of an object change over time. In mathematical terms, it is the difference between the final velocity (\(v_f\)) and the initial velocity (\(v_i\)).
For example, if a car initially moves at \(+82 \text{ m/s}\) and then at \(-82 \text{ m/s}\), the change in velocity would be:\[\Delta v = v_f - v_i = -82 \text{ m/s} - (+82 \text{ m/s}) = -164 \text{ m/s}\]
  • Change in velocity can be positive or negative.
  • A negative value indicates a reversal in direction.
  • It aids in calculating average acceleration.
Understanding change in velocity is crucial for determining how and why acceleration occurs in different scenarios.
Velocity
Velocity is a vector quantity that describes the speed of an object in a specific direction. It differs from speed in that it incorporates direction, making it essential for calculating changes in an object's motion.
For instance, a car moving at \(82 \text{ m/s}\) to the right and then with the same speed to the left demonstrates different velocities due to opposite directions. Velocity guides us in figuring out how acceleration takes place.
  • Velocity includes direction, unlike speed.
  • Neutral direction changes involve a velocity shift.
Understanding velocity enables us to effectively determine changes leading to acceleration.
Time
Time plays a fundamental role in computing average acceleration as it measures the duration over which a change in velocity takes place. The formula for average acceleration \(\[ a_{avg} = \frac{\Delta v}{\Delta t} \]\) incorporates time as a denominator. This implies that longer durations result in smaller acceleration values if the change in velocity is the same.
For instance, with \(\Delta t = 12\) seconds in the exercise, it helps in determining the rate of change in velocity.
  • Time impacts how quickly or slowly acceleration occurs.
  • Shorter time frames result in higher average acceleration,for the same change in velocity.
Thus, understanding time's role is vital for analyzing and predicting acceleration outcomes.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In 1954 the English runner Roger Bannister broke the four-minute barrier for the mile with a time of \(3 : 59.4 \mathrm{s}(3 \mathrm{min} \text { and } 59.4 \mathrm{s}) .\) In 1999 the Moroccan runner Hicham el-Guerrouj set a record of \(3 : 43.13\) s for the mile. If these two runners had run in the same race, each running the entire race at the average speed that earned him a place in the record books, el-Guerrouj would have won. By how many meters?

A car is traveling at a constant speed of 33 \(\mathrm{m} / \mathrm{s}\) on a highway. At the instant this car passes an entrance ramp, a second car enters the highway from the ramp. The second car starts from rest and has a constant acceleration. What acceleration must it maintain, so that the two cars meet for the first time at the next exit, which is 2.5 \(\mathrm{km}\) away?

A cheetah is hunting. Its prey runs for 3.0 \(\mathrm{s}\) at a constant velocity of \(+9.0 \mathrm{m} / \mathrm{s}\) . Starting from rest, what constant acceleration must the cheetah maintain in order to run the same distance as its prey runs in the same time?

A runner is at the position \(x=0 \mathrm{m}\) when time \(t=0\) s. One hundred meters away is the finish line. Every ten seconds, this runner runs half the remaining distance to the finish line. During each ten-second segment, the runner has a constant velocity. For the first forty seconds of the motion, construct (a) the position-time graph and \((b)\) the velocity-time graph.

A ball is thrown vertically upward, which is the positive direction. A little later it returns to its point of release. The ball is in the air for a total time of 8.0 s. What is its initial velocity? Neglect air resistance.
See all solutions

Recommended explanations on Physics Textbooks

Magnetism

Read Explanation

Space Physics

Read Explanation

Electricity

Read Explanation

Waves Physics

Read Explanation

Scientific Method Physics

Read Explanation

Electrostatics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.