The positions of two objects, P1 and P2, on a coordinate...

The positions of two objects, $P_{1}$ and $P_{2}$, on a coordinate line at the end of $t$ seconds are given by $s_{1}=3 t^{3}-12 t^{2}+$ $18 t+5$ and $s_{2}=-t^{3}+9 t^{2}-12 t,$ respectively. When do the two objects have the same velocity?

Key Concepts

Position Functions

A position function represents an object’s location on a coordinate line as a function of time. In kinematics, knowing the position functions allows for the analysis of motion by providing a direct relationship between an object's displacement and time.

Differentiation

Differentiation is the process used in calculus to determine the rate of change of a function with respect to a variable. It is a fundamental tool in physics and mathematics for finding derivatives, which are essential in understanding how quantities change over time.

Velocity Functions

A velocity function is obtained by differentiating a position function with respect to time. It quantifies the speed and direction of an object’s motion, serving as a critical concept in understanding dynamics in one-dimensional motion.

Equating Derivatives

When two objects are moving, comparing their velocities involves setting their corresponding velocity functions equal to each other. By solving the resulting equation, one can determine the times at which the objects have the same velocity.

Solving Polynomial Equations

In many kinematics problems, the position and velocity functions are expressed as polynomial functions. Solving the equations that result from setting two polynomial functions equal typically involves finding the roots of the polynomial, which offers the key time values where certain conditions, such as equal velocity, are met.

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