The position of an object is given by 𝓍 = At + Bt², where 𝓍 is in meters and t is in seconds.                                                                                                         
                                                                                                                                                                                          
(b) What is the acceleration as a function of time?"

Verified step by step guidance

1. Identify the given position function, which is x = At + Bt^2.
2. Recall that velocity is the first derivative of position with respect to time. Differentiate the position function x = At + Bt^2 with respect to time t to find the velocity function v(t).
3. Apply the power rule of differentiation: The derivative of t^n with respect to t is n*t^(n-1).
4. Recall that acceleration is the derivative of velocity with respect to time. Differentiate the velocity function v(t) you obtained in step 2 to find the acceleration function a(t).
5. Apply the power rule of differentiation again to find the derivative of the velocity function, which will give you the acceleration as a function of time.

Key Concepts

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. They relate position, velocity, acceleration, and time, allowing us to predict an object's future position or velocity based on its current state. The equation given, 𝓍 = At + Bt², is a polynomial representation of motion, where A and B are coefficients that define the linear and quadratic components of the motion.

Acceleration

Acceleration is the rate of change of velocity with respect to time. In the context of the given equation, acceleration can be derived by taking the second derivative of the position function with respect to time. This means that acceleration is constant if the coefficients A and B are constants, and it can be expressed as a function of time based on the terms in the position equation.

Derivatives in Physics

Derivatives are fundamental in physics for analyzing how quantities change. The first derivative of the position function gives velocity, while the second derivative provides acceleration. Understanding how to apply derivatives to motion equations is crucial for determining how an object's speed and direction change over time, which is essential for solving problems related to motion.