An object moves along a line so that its position s relative...

An object moves along a line so that its position s relative to a starting point at any time $t \geqslant 0$ is given by $s(t)=\cos \left(t^{2}-1\right)$ a) Find the velocity of the object as a function of $t$ b) What is the object's velocity at $t=0 ?$ c) In the interval $0 < t < 2.5,$ find any times (values of $t$ ) for which the object is stationary. d) Describe the object's motion during the interval $0 < t < 2.5$

An object moves along a line so that its position s relative to a starting point at any time $t \geqslant 0$ is given by $s(t)=\cos \left(t^{2}-1\right)$
a) Find the velocity of the object as a function of $t$
b) What is the object's velocity at $t=0 ?$
c) In the interval $0 < t < 2.5,$ find any times (values of $t$ ) for which the object is stationary.
d) Describe the object's motion during the interval $0 < t < 2.5$

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Key Concepts

Stationary Points

Stationary points occur when the derivative of a function is zero, indicating that the rate of change is temporarily halted. In motion analysis, finding where the velocity (the derivative of position) is zero helps determine the moments when the object is at rest or changing the direction of motion.

Derivative

The derivative represents the instantaneous rate of change of a function with respect to a variable. In the context of kinematics, the derivative of the position function with respect to time gives the velocity, which describes how fast the object's position changes at any given moment.

Chain Rule

The chain rule is a fundamental differentiation technique used to compute the derivative of composite functions. When the position function is given as a composite function (such as applying a trigonometric function to a quadratic expression), the chain rule is employed to correctly differentiate the function.

Motion Analysis

In kinematics and calculus, analyzing the sign and behavior of the velocity function provides insight into the object’s movement. This includes determining intervals of motion, points of rest, and changes in direction, which collectively describe the object's overall motion during the given time interval.

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