Question

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

Short answer

Answer: The instantaneous velocity function is \(v(t) = 4 A t^{3} - 3 B t^{2}\), and the instantaneous acceleration function is \(a(t) = 12 A t^{2} - 6 B t\).

Step-by-step solution

Differentiate the position function to find the velocity function

To find the instantaneous velocity as a function of time, we need to differentiate the position function with respect to time. So, we will differentiate \(x=A t^{4}-B t^{3}+C\) with respect to \(t\). Using the power rule of differentiation, we have: \(v(t) = \frac{dx}{dt} = 4 A t^{3} - 3 B t^{2}\)

Differentiate the velocity function to find the acceleration function

Now that we have the velocity function, we can find the instantaneous acceleration as a function of time by differentiating the velocity function with respect to time. So, we will differentiate \(v(t) = 4 A t^{3} - 3 B t^{2}\) with respect to \(t\). Using the power rule of differentiation again, we have: \(a(t) = \frac{dv}{dt} = 12 A t^{2} - 6 B t\)

Final answers

a) The instantaneous velocity as a function of time is \(v(t) = 4 A t^{3} - 3 B t^{2}\). b) The instantaneous acceleration as a function of time is \(a(t) = 12 A t^{2} - 6 B t\).

Key concepts

Instantaneous Velocity

Understanding the concept of instantaneous velocity is crucial when studying motion. It represents the speed and direction of an object at a specific point in time. Unlike average velocity, which considers the total displacement over a total time period, instantaneous velocity is focused on an infinitesimally small interval of time, giving a real-time speed.

To grasp this, imagine you're watching a car race. The racer's speed meter at any given moment reflects the car's instantaneous velocity. It tells you precisely how fast the car is moving at that exact second, which could be different from the car's average speed over several laps.

In mathematical terms, to find the instantaneous velocity, we differentiate the position function with respect to time. This is because differentiation gives us the rate at which a quantity changes, which is exactly what velocity measures - the rate of change of position.

Power Rule of Differentiation

The power rule is a fundamental technique in calculus used to find the derivative of a function where the variable has an exponent. This rule greatly simplifies the process of differentiation when dealing with polynomials or any function that can be expressed as a power of the variable.

The rule is straightforward: for any term with the form \( x^n \), where \( n \) is a real number, the derivative is \( n \( x^{n-1} \) \). For example, the derivative of \( x^4 \) is \( 4x^3 \), and the derivative of \( x^3 \) is \( 3x^2 \).

Applying the power rule not only yields the instantaneous velocity when differentiating the position function but also helps us derive the equation for instantaneous acceleration by differentiating the velocity function with respect to time. It's a critical tool for physics problems involving motion.

Position Function

The position function, often denoted as \( x(t) \) or \( s(t) \) in physics and mathematics, precisely describes an object's location relative to a reference point, at any given time. This function is fundamental in kinematics as it lays the groundwork for understanding motion.

For example, the given position function \( x=A t^{4}-B t^{3}+C \) encapsulates how the object's position changes in relation to time. Such an equation may seem complex, but it smoothly translates an object's motion into an algebraic expression that can be manipulated and analyzed to extract meaningful information about the object's behavior over time, such as its velocity and acceleration.

It's important to realize that the position function is the starting point in solving kinematics problems. Without it, determining the instantaneous velocity or acceleration would not be possible.