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 66

Chapter 2, Problem 66

A motorist is driving at 20 m/s when she sees that a traffic light 200 m ahead has just turned red. She knows that this light stays red for 15 s, and she wants to reach the light just as it turns green again. It takes her 1.0 s to step on the brakes and begin slowing. What is her speed as she reaches the light at the instant it turns green?

Verified step by step guidance

Step 1: Determine the total time available for the motorist to reach the traffic light. The light stays red for 15 seconds, and the motorist takes 1 second to react before braking. Therefore, the total time available for her to reach the light is 15 seconds.

Step 2: Calculate the distance the motorist travels during her reaction time (1 second) while maintaining her initial speed of 20 m/s. Use the formula for distance traveled at constant speed:

d = v \times t

, where

v

is the speed and

t

is the time.

Step 3: Subtract the distance traveled during the reaction time from the total distance to the traffic light (200 m). This gives the remaining distance the motorist needs to cover while decelerating.

Step 4: Use the kinematic equation

d = v

₀t+12at² to relate the remaining distance, initial speed, acceleration, and time. Here,

v

₀ is the initial speed (20 m/s),

a

is the acceleration (to be determined), and

t

is the time available for deceleration (14 seconds, after subtracting the reaction time). Solve for the acceleration.

Step 5: Use the kinematic equation

v = v

₀+at to calculate the final speed of the motorist as she reaches the traffic light. Substitute the values for initial speed, acceleration, and time to find the final speed.

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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 motorist's motion as she decelerates and to calculate her final speed when she reaches the traffic light.

Acceleration

Acceleration is 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 motorist will experience negative acceleration as she applies the brakes, which is crucial for determining how her speed changes over the distance to the traffic light.

Time Management in Motion

Time management in motion refers to how time intervals affect the position and speed of moving objects. In this case, the motorist has a total of 15 seconds before the light turns green, but she takes 1 second to react before braking. Understanding how to allocate this time effectively is key to calculating her speed at the moment the light turns green.

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