A particle moves along x -axis in such a way that its...

A particle moves along $x$ -axis in such a way that its position coordinate $(x)$ varies with time $(t)$ according to the expression $x=2-5 t+6 t^{2}$. Its initial velocity is a. $-3 \mathrm{~m} / \mathrm{s}$ b. $-5 \mathrm{~m} / \mathrm{s}$ c. $2 \mathrm{~m} / \mathrm{s}$ d. $3 \mathrm{~m} / \mathrm{s}$

A particle moves along $x$ -axis in such a way that its position coordinate $(x)$ varies with time $(t)$ according to the expression $x=2-5 t+6 t^{2}$. Its initial velocity is
a. $-3 \mathrm{~m} / \mathrm{s}$
b. $-5 \mathrm{~m} / \mathrm{s}$
c. $2 \mathrm{~m} / \mathrm{s}$
d. $3 \mathrm{~m} / \mathrm{s}$

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Key Concepts

Kinematics in One Dimension

This concept involves analyzing the motion of a particle that moves along a straight line. It includes understanding how the position of the particle changes over time and how this change relates to its velocity and acceleration. In one-dimensional motion, the position is described by a function of time, and the motion is characterized by its initial conditions and the forces acting on the particle.

Differentiation to Determine Velocity

Velocity is defined as the rate of change of position with respect to time. In problems involving motion, taking the derivative of the position function with respect to time yields the velocity function. This application of differentiation is a fundamental tool in calculus that allows for the analysis of how rapidly and in which direction the particle is moving at any given moment.

Initial Conditions in Motion Analysis

The concept of initial conditions refers to the state of a system at the beginning of an observation, typically at time zero. In kinematics, knowing the initial position, velocity, and sometimes acceleration of a particle is crucial for predicting its future motion. Evaluating the velocity function at time zero gives the initial velocity, which is a key parameter in solving motion problems.

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