At the instant the traffic light turns green, a car that has been waiting at an intersection starts ahead with a constant acceleration of 2.802.80 m/s2. At the same instant a truck, traveling with a constant speed of 20.020.0 m/s, overtakes and passes the car. How far beyond its starting point does the car overtake the truck?

Ch 02: Motion Along a Straight Line

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 02: Motion Along a Straight Line

Problem 34a

Chapter 2, Problem 34a

At the instant the traffic light turns green, a car that has been waiting at an intersection starts ahead with a constant acceleration of $2.80$ m/s². At the same instant a truck, traveling with a constant speed of $20.0$ m/s, overtakes and passes the car. How far beyond its starting point does the car overtake the truck?

Verified step by step guidance

First, let's define the motion equations for both the car and the truck. The car starts from rest and accelerates with a constant acceleration, so its position as a function of time can be described by the equation:

x = \frac{1}{2} a t^{2}

, where

a

is the acceleration and

t

is the time.

For the truck, which is moving at a constant speed, its position as a function of time is given by:

x = v t

, where

v

is the constant speed of the truck.

To find the point where the car overtakes the truck, set the position equations equal to each other:

\frac{1}{2} a t^{2} = v t

. This equation will allow us to solve for the time

t

when the car overtakes the truck.

Rearrange the equation to solve for

t

t^{2} = \frac{2 v t}{a}

. Then, solve for

t

by taking the square root of both sides.

Once you have the value of

t

, substitute it back into the car's position equation

x = \frac{1}{2} a t^{2}

to find the distance beyond the starting point where the car overtakes the truck.

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Key Concepts

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

Constant Acceleration

Constant acceleration refers to a situation where an object's velocity changes at a steady rate over time. In this problem, the car accelerates at 2.80 m/s², meaning its speed increases by 2.80 meters per second every second. This concept is crucial for calculating the car's position over time using kinematic equations.

Constant Speed

Constant speed means an object moves at a uniform rate, covering equal distances in equal time intervals. The truck travels at a constant speed of 20.0 m/s, which simplifies its motion analysis since its position can be calculated directly by multiplying speed by time, without needing to account for acceleration.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. They relate displacement, initial velocity, final velocity, acceleration, and time. For the car, these equations help determine its position as it accelerates, allowing us to find the point where it overtakes the truck by equating the positions of both vehicles over time.

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Textbook Question

A cat walks in a straight line, which we shall call the $x$ -axis, with the positive direction to the right. As an observant physicist, you make measurements of this cat's motion and construct a graph of the feline's velocity as a function of time (Fig. E $2.30$ ). Assuming that the cat started at the origin, sketch clear graphs of the cat's acceleration and position as functions of time.

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Textbook Question

At the instant the traffic light turns green, a car that has been waiting at an intersection starts ahead with a constant acceleration of $2.80$ m/s². At the same instant a truck, traveling with a constant speed of $20.0$ m/s, overtakes and passes the car. How fast is the car traveling when it overtakes the truck?

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