Find an expression for the minimum stopping distance dstop of a car traveling at speed v0 if the driver's reaction time is Treact and the magnitude of the acceleration during maximum braking is a constant abrake.

Ch 02: Kinematics in One Dimension

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 02: Kinematics in One Dimension

Problem 53a

Chapter 2, Problem 53a

Find an expression for the minimum stopping distance d_stop of a car traveling at speed v₀ if the driver's reaction time is T_react and the magnitude of the acceleration during maximum braking is a constant a_brake.

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Identify the two phases of the car's motion: (1) the reaction phase, where the car travels at constant speed during the driver's reaction time, and (2) the braking phase, where the car decelerates uniformly to a stop.

For the reaction phase, calculate the distance traveled during the reaction time \( T_{\text{react}} \) using the formula \( d_{\text{react}} = v_0 \cdot T_{\text{react}} \), where \( v_0 \) is the initial speed of the car.

For the braking phase, use the kinematic equation \( v_f^2 = v_0^2 + 2a_{\text{brake}}d_{\text{brake}} \), where \( v_f = 0 \) (final velocity), \( a_{\text{brake}} \) is the constant deceleration, and \( d_{\text{brake}} \) is the stopping distance during braking. Rearrange to find \( d_{\text{brake}} = \frac{-v_0^2}{2a_{\text{brake}}} \).

Combine the distances from both phases to find the total stopping distance: \( d_{\text{stop}} = d_{\text{react}} + d_{\text{brake}} \). Substituting the expressions, \( d_{\text{stop}} = v_0 \cdot T_{\text{react}} + \frac{-v_0^2}{2a_{\text{brake}}} \).

Simplify the expression to obtain the final formula for the minimum stopping distance: \( d_{\text{stop}} = v_0 \cdot T_{\text{react}} + \frac{v_0^2}{2a_{\text{brake}}} \). This formula accounts for both the reaction time and the braking phase.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the forces that cause the motion. It involves concepts such as displacement, velocity, and acceleration. In this context, understanding kinematics is essential to analyze how a car's speed changes over time, particularly during braking.

Acceleration

Acceleration is the rate of change of velocity of an object with respect to time. It can be positive (speeding up) or negative (slowing down, also known as deceleration). In the problem, the constant acceleration during maximum braking, denoted as a₆ᵣₐₖₑ, is crucial for determining how quickly the car can stop from its initial speed v₀.

Reaction Time

Reaction time is the delay between the perception of a stimulus and the initiation of a response. In driving, it refers to the time it takes for a driver to react to a situation, such as pressing the brakes. The driver's reaction time, Tᵣₑₐ꜀ₜ, affects the total stopping distance, as it contributes to the distance traveled before braking begins.

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