A car is stopped at a traffic light. It then travels along a straight road such that its distance from the light is given by x(t)=bt2−ct3x(t)=$$bt^2$-ct^3$$, where b=2.40b = 2.40 m/s2 and c=0.120c = 0.120 m/s3. Calculate the average velocity of the car for the time interval t=0t = 0 to t=10.0t = 10.0 s.

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 7a

Chapter 2, Problem 7a

A car is stopped at a traffic light. It then travels along a straight road such that its distance from the light is given by $x (t) = b t^{2} - c t^{3}$ , where $b equals 2.40$ m/s² and $c equals 0.120$ m/s³. Calculate the average velocity of the car for the time interval $t equals 0$ to $t equals 10.0$ s.

Verified step by step guidance

Understand the problem: We need to calculate the average velocity of the car over the time interval from t = 0 to t = 10.0 seconds. The position of the car as a function of time is given by the equation x(t) = bt^2 − ct^3, where b = 2.40 m/s^2 and c = 0.120 m/s^3.

Calculate the position of the car at t = 0 seconds: Substitute t = 0 into the equation x(t) = bt^2 − ct^3 to find the initial position x(0). This will help us determine the change in position over the time interval.

Calculate the position of the car at t = 10.0 seconds: Substitute t = 10.0 into the equation x(t) = bt^2 − ct^3 to find the final position x(10). This will give us the position of the car at the end of the time interval.

Determine the change in position: Subtract the initial position x(0) from the final position x(10) to find the total displacement of the car over the time interval.

Calculate the average velocity: Use the formula for average velocity, which is the total displacement divided by the total time interval. The average velocity v_avg is given by v_avg = (x(10) - x(0)) / (10.0 - 0).

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Key Concepts

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

Average Velocity

Average velocity is defined as the total displacement divided by the total time taken. It is a vector quantity that considers the direction of motion. In this problem, it involves calculating the change in position of the car from t = 0 to t = 10 seconds and dividing by the time interval.

Displacement Function

The displacement function x(t) = bt² − ct³ describes the car's position as a function of time. It is crucial to understand how to evaluate this function at specific time points to determine the car's position at those times. This function involves quadratic and cubic terms, indicating non-linear motion.

Integration of Motion Equations

Understanding how to integrate motion equations is essential for solving problems involving variable acceleration. The given function x(t) can be differentiated to find velocity and acceleration, but for average velocity, evaluating the function at specific times suffices. This concept helps in understanding the relationship between position, velocity, and acceleration.

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