A particle moves with acceleration function a(t)=5+4 t-2 t^2...

A particle moves with acceleration function $a(t)=5+4 t-2 t^{2} .$ Its initial velocity is $v(0)=3 \mathrm{m} / \mathrm{s}$ and its initial displacement is $s(0)=10 \mathrm{m} .$ Find its position after $t$ seconds.

Key Concepts

Initial Conditions

Initial conditions, such as initial velocity and initial displacement, provide specific values that are used to determine the constants arising from the integration process. These conditions ensure that the calculated functions for velocity and displacement accurately reflect the particle's motion from the given starting point.

Displacement (Position)

Displacement, or position, is a function that shows the location of a particle as a function of time. It is obtained by integrating the velocity function. The inclusion of the initial displacement value is crucial to determine the constant of integration, thus providing the specific trajectory of the object's motion.

Integration

Integration is the mathematical process of finding the antiderivative of a function. In the context of motion, integrating the acceleration function yields the velocity function, and further integrating the velocity function provides the displacement function. This method is essential for solving problems where initial conditions are given to find unique solutions for motion.

Acceleration

Acceleration is the rate at which an object’s velocity changes with time. It is defined mathematically as the derivative of velocity with respect to time and serves as the starting point for problems involving motion, as it directly influences how velocity and subsequently displacement change over time.

Velocity

Velocity represents the rate of change of displacement with respect to time, and is derived as the integral of the acceleration function. The resulting function describes how fast and in what direction an object is moving at any given time, and it incorporates any initial velocity through the constant of integration.

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