The velocity of a car (in feet/second) t sec after starting...

The velocity of a car (in feet/second) $t$ sec after starting from rest is given by the function
$$
f(t)=2 \sqrt{t} \quad(0 \leq t \leq 30)
$$
Find the car's position, $s(t)$, at any time $t$. Assume $s(0)=0$.

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Key Concepts

Initial Conditions in Differential Equations

When integrating to find a function such as the position from velocity, an arbitrary constant of integration appears. Initial conditions, such as the starting position at time zero, are used to determine this constant, ensuring that the solution satisfies the specific conditions of the problem.

Power Rule for Integration

The power rule for integration is a basic technique used to find the antiderivative of power functions. It involves increasing the exponent of the variable by one and then dividing by the new exponent, which is crucial for calculating integrals of functions that can be expressed in the form t^n.

Integration in Kinematics

In kinematics, the position of an object is determined by integrating its velocity with respect to time. This process accumulates the total displacement over a given time interval, reflecting the continuous change in position as time progresses.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the concepts of differentiation and integration. It states that if a function is continuous over an interval, its definite integral from one point to another can be computed by taking the antiderivative evaluated at those points, allowing us to determine the accumulation of quantities like displacement.

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