The acceleration a in m / s^2 of a particle is given by a=3 t^2+2 t+2 where t is the time in sec

step 1 In this problem we have a is equal to 3 t square plus 2 t plus 2 d b is equal to a d t so b is e

The acceleration a in m / s^2 of a particle is given by a=3...

The acceleration $a$ in $\mathrm{m} / \mathrm{s}^{2}$ of a particle is given by $a=3 t^{2}+2 t+2$ where $t$ is the time in sec. If the particle starts out with a velocity $v=2 \mathrm{~m} / \mathrm{s}$ at $t=0$, then find the velocity at the end of $2 \mathrm{~s}$.

Key Concepts

Acceleration

Acceleration is the rate of change of velocity with respect to time. It quantifies how quickly an object's speed or direction is changing and is fundamental in kinematics to describe motion. In many problems, acceleration is given as a function of time, which allows for further analysis of the object's velocity and displacement.

Integration of Acceleration

Since velocity is the integral of acceleration with respect to time, finding the velocity function involves integrating the acceleration function. This process of integration recovers the original function, in this case, the velocity, up to an additive constant which represents the initial velocity.

Initial Conditions

Initial conditions provide specific values for a function at a certain point in time, which are used to determine the constant of integration when integrating differential equations. In the context of kinematics, knowing the initial velocity allows one to compute the precise velocity function that fits the motion of the particle.

Definite Integration

Definite integration is used to compute the net change in a quantity over a specified interval. When integrating the acceleration function from a starting time to an ending time, the definite integral yields the total change in velocity over that interval. This method is essential for solving problems where the initial value and time interval are defined.

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