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)=$$bt^2$-ct^3x(t)=bt2$$−ct3, where b=2.40b = 2.40b=2.40 m/s2 and c=0.120c = 0.120c=0.120 m/s3. Calculate the instantaneous velocity of the car at t=0t = 0, t=5.0t = 5.0 s, and t=10.0t = 10.0 s.

Ch 02: Motion Along a Straight Line

Young & Freedman Calc - University Physics 15th Edition

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Young & Freedman Calc 15th Edition

Ch 02: Motion Along a Straight Line

Problem 7b

Chapter 2, Problem 7b

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 $x(t)=bt^2-ct^3$ , where $b = 2.40$ m/s² and $c = 0.120$ m/s³. Calculate the instantaneous velocity of the car at $t equals 0$ , $t equals 5.0$ s, and $t equals 10.0$ s.

Verified step by step guidance

To find the instantaneous velocity of the car, we need to take the derivative of the position function x(t) with respect to time t. The position function given is x(t) = bt^2 − ct^3.

The derivative of x(t) with respect to t, which gives us the velocity function v(t), is v(t) = d(x(t))/dt = d(bt^2)/dt - d(ct^3)/dt.

Calculate the derivative of each term separately: The derivative of bt^2 with respect to t is 2bt, and the derivative of ct^3 with respect to t is 3ct^2.

Combine the derivatives to get the velocity function: v(t) = 2bt - 3ct^2.

Substitute the given values of b and c into the velocity function and evaluate it at the specified times t = 0, t = 5.0 s, and t = 10.0 s to find the instantaneous velocities at these times.

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Key Concepts

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

Instantaneous Velocity

Instantaneous velocity is the speed of an object at a specific moment in time. It is determined by taking the derivative of the position function with respect to time, which gives the rate of change of position. In this problem, the position function x(t) = bt² - ct³ needs to be differentiated to find the velocity function v(t).

Differentiation

Differentiation is a mathematical process used to find the rate at which a quantity changes. In physics, it is often used to calculate velocity from a position-time function. For the given function x(t) = bt² - ct³, differentiating with respect to time t gives v(t) = d(x)/dt = 2bt - 3ct², which is essential for finding instantaneous velocities.

Position-Time Function

A position-time function describes how the position of an object changes over time. In this problem, x(t) = bt² - ct³ represents the car's position relative to the traffic light. Understanding this function is crucial for determining how the car's position evolves, which is necessary for calculating its velocity at specific times.

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Textbook Question

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Textbook Question

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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.

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