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 − ct3, where b = 2.40 m/s2 and c = 0.120 m/s3. (b) Calculate the instantaneous velocity of the car at t = 0, t = 5.0 s, and t = 10.0 s.

Verified step by step guidance

1. Identify the given function for the position of the car, x(t) = bt^2 - ct^3, where b = 2.40 m/s^2 and c = 0.120 m/s^3.
2. Calculate the derivative of x(t) with respect to time t to find the velocity function, v(t). Use the power rule for differentiation: v(t) = d/dt [bt^2 - ct^3] = 2bt - 3ct^2.
3. Substitute the values of b and c into the velocity function to simplify it: v(t) = 2(2.40)t - 3(0.120)t^2 = 4.8t - 0.36t^2.
4. Evaluate the velocity function at t = 0 s to find the instantaneous velocity at that time: v(0) = 4.8(0) - 0.36(0)^2.
5. Evaluate the velocity function at t = 5.0 s and t = 10.0 s to find the instantaneous velocities at those times: v(5.0) = 4.8(5.0) - 0.36(5.0)^2 and v(10.0) = 4.8(10.0) - 0.36(10.0)^2.

Key Concepts

Instantaneous Velocity

Instantaneous velocity is the rate of change of an object's position with respect to time at a specific moment. It is mathematically defined as the derivative of the position function x(t) with respect to time t. This concept is crucial for understanding how fast an object is moving at any given instant, as opposed to average velocity, which considers the total distance over a time interval.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. In the context of motion, differentiating the position function x(t) provides the instantaneous velocity function v(t). This process allows us to analyze how the position of an object changes over time, which is essential for solving problems related to motion.

Polynomial Functions

Polynomial functions are mathematical expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. In this problem, the position function x(t) = bt² - ct³ is a polynomial of degree three. Understanding polynomial functions is important because their derivatives can be easily computed, allowing for straightforward analysis of motion in physics.