A car is stopped at a traffic light. It then travels along a...

A car is stopped at a traffic light. It then travels along a straight road so that its distance from the light is given by $x(t)=b t^{2}-c t^{3},$ where $b=240 \mathrm{m} / \mathrm{s}^{2}$ and $c=0.120 \mathrm{m} / \mathrm{s}^{3}$ . (a) Calculate the average velocity of the car for the time interval $t=0$ to $t=10.0 \mathrm{s}$ . (b) Calculate the instantaneous velocity of the car at $t=0, t=5.0 \mathrm{s},$ and $t=10.0 \mathrm{s} .$ (c) How long after starting from rest is the car again at rest?

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Key Concepts

Solving for Stationary Points

Determining when an object is at rest involves setting the instantaneous velocity equal to zero and solving for the time variable. This process identifies the moments when the object's motion temporarily stops, which is critical in understanding various aspects of its overall motion.

Differentiation in Kinematics

Differentiation is a fundamental calculus tool used in kinematics to transition from a position function to a velocity function, and further to an acceleration function. This technique helps analyze the behavior of moving objects by mathematically describing how their motion variables change over time.

Average Velocity

Average velocity is defined as the total displacement divided by the total time taken. In kinematics, it provides a measure of how fast an object moves overall between two points in time, without detailing its speed changes during the interval.

Instantaneous Velocity

Instantaneous velocity is the velocity of an object at a specific moment in time. It is obtained by taking the derivative of the position function with respect to time, thus offering a precise measure of how fast the object is moving at that particular instant.

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