An object's position as a function of time is given by x=b...

An object's position as a function of time is given by $x=b t^{2}-c t^{4}$, where $b$ has the value $1.82 \mathrm{~m} / \mathrm{s}^{2}$, which puts the object at $x=0$ at $t=0$. (a) Find the value of $c$ such that the object will again be at $x=0$ when $t=2.54 \mathrm{~s}$. Also, find (b) the object's speed and (c) its acceleration at that time.

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Key Concepts

Solving Algebraic Equations for Unknown Parameters

In the context of kinematics, it is often necessary to determine unknown coefficients or parameters in the position function. This can involve setting the function equal to specific values (such as a known position at a certain time) and solving the resulting algebraic equation to ensure the function adheres to given boundary conditions.

Differentiation to Determine Acceleration

Acceleration is the second derivative of the position function or the first derivative of the velocity function. It represents the rate at which the velocity of the object changes over time, offering insights into how forces may be acting on the object.

Position Function in Kinematics

This concept involves describing an object's location as a function of time using a mathematical expression. The position function encapsulates the motion of the object and serves as the starting point for further analysis of its motion, such as determining velocity and acceleration through differentiation.

Differentiation to Determine Velocity

Velocity is the first derivative of the position function with respect to time. Calculating the derivative of the position function provides the instantaneous rate of change of position, which reflects the speed and direction of the object’s motion at any given time.

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