At the instant the traffic light turns green, a car that has been waiting at an intersection starts ahead with a constant acceleration of 2.802.802.80 m/s2. At the same instant a truck, traveling with a constant speed of 20.020.020.0 m/s, overtakes and passes the car. How fast is the car traveling when it overtakes 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 34b

Chapter 2, Problem 34b

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?

Verified step by step guidance

Identify the equations of motion for both the car and the truck. The car starts from rest with a constant acceleration, so its velocity as a function of time is given by:

v = a t

, where

a

is the acceleration and

t

is the time. The truck travels at a constant speed, so its velocity is

v = 20.0 m/s

Determine the position of the car as a function of time using the equation:

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

. Since the car starts from rest, its initial position is zero.

Determine the position of the truck as a function of time using the equation:

x = v t

, where

v

is the constant speed of the truck.

Set the position equations of the car and the truck equal to each other to find the time at which the car overtakes the truck:

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

. Solve this equation for

t

Substitute the time found in the previous step into the car's velocity equation

v = a t

to find the speed of the car when it 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. Understanding this concept is crucial for calculating the car's velocity at any given time.

Relative Motion

Relative motion involves analyzing the movement of one object concerning another. Here, the car and truck are moving relative to each other, with the truck initially overtaking the car. To find when the car overtakes the truck, we must consider their relative speeds and positions over time.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. They relate displacement, initial velocity, final velocity, acceleration, and time. In this scenario, these equations help determine the car's velocity when it catches up to the truck, using its initial velocity, acceleration, and the time taken to overtake.

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