Ch 02: Kinematics in One Dimension

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 02: Kinematics in One Dimension

Problem 47

Chapter 2, Problem 47

You are driving to the grocery store at 20 m/s. You are 110 m from an intersection when the traffic light turns red. Assume that your reaction time is 0.50 s and that your car brakes with constant acceleration. What magnitude braking acceleration will bring you to a stop exactly at the intersection?

Verified step by step guidance

Determine the distance traveled during the reaction time. Use the formula for distance traveled at constant velocity:

d = v t

, where

v

is the velocity (20 m/s) and

t

is the reaction time (0.50 s).

Subtract the distance traveled during the reaction time from the total distance to the intersection to find the distance over which the car must decelerate. Use the formula:

d = d total - d reaction

, where

d total

is 110 m.

Use the kinematic equation to relate the remaining distance, initial velocity, final velocity, and acceleration:

d = v i t + \frac{1}{2} a t^{2}

. Here,

v i

is the initial velocity (20 m/s),

v f

is the final velocity (0 m/s), and

d

is the remaining distance.

Rearrange the kinematic equation to solve for the acceleration

a

. Use the formula:

a = \frac{- v^{i 2}}{2 d}

, where

d

is the remaining distance and

v i

is the initial velocity.

Substitute the known values into the formula for acceleration and simplify to find the magnitude of the braking acceleration. Remember to take the absolute value of the result since the problem asks for the magnitude.

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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 mechanics that deals with the motion of objects without considering the forces that cause the motion. It involves concepts such as displacement, velocity, and acceleration. In this scenario, understanding kinematics is essential to analyze the car's motion as it approaches the intersection and to calculate the necessary braking distance.

Acceleration

Acceleration is defined as the rate of change of velocity of an object over time. It can be positive (speeding up) or negative (slowing down, also known as deceleration). In this problem, the braking acceleration is crucial to determine how quickly the car can stop before reaching the intersection, given the initial speed and the distance to the stop.

Reaction Time

Reaction time is the delay between the perception of a stimulus and the initiation of a response. In driving scenarios, it affects how quickly a driver can respond to changes, such as a traffic light turning red. In this question, the 0.50 s reaction time must be accounted for when calculating the total distance traveled before the brakes are applied, impacting the required braking acceleration.

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Textbook Question

FIGURE P2.45 shows a set of kinematic graphs for a ball rolling on a track. All segments of the track are straight lines, but some may be tilted. Draw a picture of the track and also indicate the ball's initial condition.

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Textbook Question

You're driving down the highway late one night at 20 m/s when a deer steps onto the road 35 m in front of you. Your reaction time before stepping on the brakes is 0.50 s, and the maximum deceleration of your car is 10 m/s². How much distance is between you and the deer when you come to a stop?

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Textbook Question

A car starts from rest at a stop sign. It accelerates at 4.0 m/s² for 6.0 s, coasts for 2.0 s, and then slows down at a rate of 3.0 m/s² for the next stop sign. How far apart are the stop signs?

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Textbook Question

A particle's velocity is given by the function $$\mathcal{v}$_x = (2.0 \, $\text{m/s}$) $\sin$($\pi$ t)$ , where $t$ is in $s$ . What is the particle's acceleration at that time?

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