Question

The position of an object as a function of time is given by \(x=A t^{4}-B t^{3}+C\). a) What is the instantaneous velocity as a function of time? b) What is the instantaneous acceleration as a function of time?

Short answer

Question: Given the position function \(x(t) = A t^{4} - B t^{3} + C\), find the functions for a) instantaneous velocity and b) instantaneous acceleration. Answer: a) The instantaneous velocity function is \(v(t) = 4At^{3} - 3Bt^{2}\) and b) the instantaneous acceleration function is \(a(t) = 12At^{2} - 6Bt\).

Step-by-step solution

a) Instantaneous Velocity Function

To find the instantaneous velocity, v(t), we need to differentiate the position function, x(t) with respect to time t. The position function is given as \(x(t) = A t^{4} - B t^{3} + C\). Differentiate x(t) with respect to t: \(v(t) = \frac{dx}{dt} = \frac{d}{dt} (A t^{4} - B t^{3} + C)\) Using the power rule of differentiation, we can calculate the derivative: \(v(t) = 4At^{3} - 3Bt^{2}\) The instantaneous velocity function as a function of time is \(v(t) = 4At^{3} - 3Bt^{2}\).

b) Instantaneous Acceleration Function

To find the instantaneous acceleration, a(t), we need to differentiate the velocity function, v(t) with respect to time t. The velocity function is given as \(v(t) = 4At^{3} - 3Bt^{2}\). Differentiate v(t) with respect to t: \(a(t) = \frac{dv}{dt} = \frac{d}{dt} (4At^{3} - 3Bt^{2})\) Using the power rule of differentiation, we can calculate the derivative: \(a(t) = 12At^{2} - 6Bt\) The instantaneous acceleration function as a function of time is \(a(t) = 12At^{2} - 6Bt\).

Key concepts

Kinematics in Physics

Kinematics is the branch of classical mechanics that describes the motion of points, objects, and systems of bodies without considering the forces that cause them to move. It focuses on displacement, velocity, and acceleration, as well as the relationships between these vector quantities. Kinematics plays a fundamental role in physics as it helps us understand how things move over time.

For example, consider an object with a given position function as a function of time, such as the one provided in the exercise. By studying kinematics, we learn to describe its motion through various physical quantities. Here, the position function is a cubic equation, which immediately suggests that the motion is not constant and is subject to change in different phases of time, as implied by the varying powers of time in the equation.

Power Rule of Differentiation

The power rule is a basic rule in calculus that is used to differentiate functions of the form f(x) = xⁿ, where n is a real number. The rule states that the derivative of f(x) with respect to x is nf(x) = nx^n-1.

This rule simplifies the process of finding the rate of change of functions—such as finding the velocity and acceleration from the position function in the given exercise. When applied to each term of the polynomial position function x(t) = At⁴ - Bt³ + C, it allows students to easily calculate the instantaneous velocity and acceleration by reducing the exponent of each t term by one and multiplying by the original exponent.

Velocity-Time Relationship

The relationship between velocity and time is crucial in understanding motion. Velocity is the rate of change of position with respect to time. Instantaneous velocity is the velocity of an object at a specific moment in time and can be found by taking the derivative of the position function with respect to time.

In the exercise, the derivative of the position function x(t) using the power rule gives the velocity function v(t) = 4At³ - 3Bt². This provides a connection between how fast an object is moving (velocity) and the time at which the measurement is taken. The relationship is depicted in a velocity-time graph where the slope at any point gives the instantaneous velocity of the object.

Acceleration-Time Relationship

Acceleration is defined as the rate of change of velocity with respect to time. It indicates how quickly an object's velocity is changing. The acceleration-time relationship can be visualized on a graph, where the slope at any point represents the instantaneous acceleration.

To find the instantaneous acceleration from the exercise, we differentiated the velocity function v(t) with respect to time. The power rule simplifies this process, yielding a(t) = 12At² - 6Bt. This function describes how the object's acceleration changes over time, providing insight into whether the object is speeding up, slowing down, or maintaining a constant acceleration.