The velocity (in feet per second) of a maglev is v(t)=0.2 t+3 (0 ≤t ≤120) At t=0, it is at the

step 1 When they give you a velocity problem, and we have 0 .2t plus 3, the key to finding its position

The velocity (in feet per second) of a maglev is v(t)=0.2...

The velocity (in feet per second) of a maglev is
$$v(t)=0.2 t+3 \quad(0 \leq t \leq 120)$$ At $t=0,$ it is at the station. Find the function giving the position of the maglev at time $t,$ assuming that the motion takes place along a straight stretch of track.

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Key Concepts

Integration as a Tool to Recover Position

In kinematics, when the velocity function of an object is known, the position function can be obtained by integrating the velocity function with respect to time. This process accumulates the total displacement over time, reflecting how distance builds as the object moves.

Constant of Integration and Initial Conditions

The integration process introduces an arbitrary constant known as the constant of integration. Determining this constant requires the use of initial conditions, which in kinematics typically include the initial position of the object. By applying the given starting value, the precise position function is obtained.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges differentiation and integration by stating that integration is the inverse process of differentiation. In the context of kinematics, differentiating the obtained position function should return the original velocity function, confirming the accuracy of the integration.

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