A 745 i BMW car can brake to a stop in a distance of 121 ft. from a speed of 60.0 mi / h . To br

step 1 In this problem, it is given that stopping distance is 121 feet. If initial speed is 60 mile per

A 745 i BMW car can brake to a stop in a distance of 121 ft....

A 745 i BMW car can brake to a stop in a distance of $121 \mathrm{ft}$. from a speed of $60.0 \mathrm{mi} / \mathrm{h} .$ To brake to a stop from a speed of $80.0 \mathrm{mi} / \mathrm{h}$ requires a stopping distance of $211 \mathrm{ft}$ What is the average braking acceleration for (a) $60 \mathrm{mi} / \mathrm{h}$ to rest, (b) $80 \mathrm{mi} / \mathrm{h}$ to rest, $(\mathrm{c}) 80 \mathrm{mi} / \mathrm{h}$ to $60 \mathrm{mi} / \mathrm{h} ?$ Express the answers in $\mathrm{mi} / \mathrm{h} / \mathrm{s}$ and in $\mathrm{m} / \mathrm{s}^{2}$ .

Key Concepts

Uniformly Accelerated Motion

This concept involves the use of kinematic equations that relate displacement, velocity, acceleration, and time under the assumption of constant acceleration. In braking scenarios, these equations (such as v² = v?² + 2a?x) are essential for determining how a vehicle decelerates over a given distance or to a particular speed, allowing one to calculate the average acceleration or deceleration when braking.

Deceleration

Deceleration is a form of acceleration that occurs when an object reduces its velocity. In vehicle braking problems, deceleration is treated as a negative acceleration. Understanding deceleration is crucial for analyzing how rapidly a vehicle can come to a stop and for ensuring that the calculations of stopping distances are accurate within a given time frame.

Average Acceleration Calculation

Average acceleration is defined as the change in velocity divided by the time taken for that change. In scenarios where an object slows down or speeds up uniformly, calculating the average acceleration can be done using relationships between velocity and displacement, eliminating the need to directly measure the time interval. This is particularly useful in braking problems where stopping distances are given.

Unit Conversion

Unit conversion is a critical step in solving physics problems to ensure consistency in the equations. When given speeds in miles per hour (mi/h) and distances in feet or meters, it is necessary to convert these units to compatible ones (such as converting mi/h to m/s and feet to meters) in order to accurately compute acceleration in standard units like m/s². This ensures that the results are both accurate and interpretable across different measurement systems.

Quadratic Relationship between Speed and Stopping Distance

The stopping distance of a vehicle under constant deceleration is proportional to the square of its initial speed, as expressed in the kinematic equation where stopping distance equals v?² divided by twice the magnitude of the acceleration. This quadratic dependence is important because it highlights how even a moderate increase in initial speed can lead to a substantially longer stopping distance, an insight critical for both vehicle safety analysis and traffic engineering.

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